 Greetings, we are getting close to the point that we will soon have the Breitwigner relation for resonances and collisions and specifically we will be discussing how the phase shifts behave at a resonance. So I will also introduce what exactly do we mean by a resonance, what are the different kinds of resonances that we talk about and so on. So in our previous class we considered scattering by a square well and our analysis was based on the continuity of the wave function and the derivative at the boundary. So that was contained in this logarithmic derivative. So this gamma we have used in our mathematical expressions so I will be using it in today's class as well and in terms of this gamma which comes here as one of the terms in the denominator which we defined on the previous slide. The scattering phase shift we wrote as a product of these two factors one which is in this blue box and the other in this rectangular bracket. So this separation we expressed in our last class. Now what we also do is to introduce a function xi of k such that e to the 2i xi is given by this ratio minus h2 over h1, h2 is the angle function of the second kind and h1 is the angle function of the first kind. So the ratio of these two angle functions with a negative sign is what gives us this ratio, this ratio is then indicated in terms of this angle xi and the phase shift is now separated in two factors one of which is this exponential function of modulus 1. Now if you look at this and you set up the you write the angle functions angle function of the first kind and angle function of the second time in terms of the Bessel function and the Neumann function because j plus i and j minus i are the angle functions of first and second kind respectively. So you can see that this xi is then nothing but the tan inverse of j over n the ratio of the Bessel function to the Neumann function and what is this quantity tan inverse j over n is nothing but the phase shift coming from the heart sphere. So the tangent of the heart sphere scattering so this delta is different from this delta. This delta is the scattering phase shift due to the square well potential that we are talking about which is how we have expressed it. One part of it which is the first box in this blue box this part which we have written as e to the 2i xi this is exactly the same as the scattering phase shift which comes from the heart sphere. So this delta is only the heart sphere scattering I have used the same symbol because on slide 66 this ratio represented the tangent of the phase shift due to a heart sphere impenetrable sphere which has got an infinite barrier. So that scattering phase shift was given by this ratio and this ratio expresses the tangent of this phase shift xi which is coming as one of the factors. The net phase shift now this is the real phase shift due to the square well potential. This phase shift can now be written as a sum of two parts one coming from what is in this blue box which is the heart sphere component as we shall call it and then there is a remaining part which will depend which will be very sensitive to the details of the potential and other parameters of the scattering problem. So what we have essentially done is to factor out a component which we shall refer to as a heart sphere component. So mind you the actual potential is not a heart sphere the heart sphere results in a total phase shift scattering phase shift one part of which is effectively the same as if it is coming from heart sphere scattering and then there is a residual part. So our interest will be very much in the residual part so this is the remaining part of the phase shift. So the 2i xi is one part of 2i delta so xi is one part of delta and then there is another part which is dependent to all the collision details of the real potential and the primary reason of course is this logarithmic derivative which appears in this term. So we introduce a complex number r plus is now this is going to be different for every partial wave there are infinite partial waves so we write it specifically for each partial wave so there is rl plus isl but for every partial wave we have a complex number r plus is which is the ratio of the derivative of the high angle function 1 to the angle function 1 multiplied by k. So this is the complex number which we introduce and r is now the real part of this complex number s is this imaginary part of this complex number that is how we have defined it. And now we have this 2i delta l this is the phase shift delta is the scattering phase shift which is now written in terms of this heart sphere component xi and another component and because I have now introduced this r and s I can write these ratios in terms of r and s instead of the angle functions because here you have defined the r and s in terms of the angle functions. So you can invert the relations and write the angle functions in terms of r and s and this is what it turns out to be and you have a very nice relationship emerging from this simplification that the scattering phase shift delta you can see is now written as a product of e to the 2i xi and another factor which is in this rectangular box and this rectangular bracket you can see is also a modulus 1. So you can also write it as e to the 2i rho or something which will be the other part of the scattering phase shift. So let us see that so that other part will be rho and that is coming from this rectangular box. So I will write this relation explicitly over here. So we have got this e to the 2i delta l at the top of this slide this second factor is of modulus 1 and we now write the second factor as e to the 2i rho because it is of modulus 1. So rho is another phase shift now rho is another angle. So the net phase shift will be twice delta is equal to twice xi plus twice rho or delta is equal to xi plus rho. So that is a relation which is emerging for every partial wave for every lth partial wave. So delta we have written as a sum of these two phase shifts. So this part is coming from the heart sphere and it does not matter the details of the potential then do not matter because it is just the heart sphere component. The second component rho is what contains the detail dynamics of the collision process. So rho is the one which will determine which will be determined by the details of the potential V. So here we are when it depends specifically on the details of the potential rho will be very sensitive to this potential it will be very sensitive to certain dynamical factors which are involved in the collision process and at some particular energies which are which we shall refer to as a resonance energies at these energies the phase shift rho this is the part other than the heart sphere components. So this will change very rapidly in the vicinity of the resonance. So it will be extremely sensitive to the resonance phenomenon and this will change very rapidly and you will see that it changes very rapidly through pi by 2 but if you consider a little bit if you go to slightly lower to lower energies and go well above the resonance then over this region then the phase shift rho or the consequent phase shift delta will change through pi. So you will see I will show these figures and that will make this whole thing very clear. So here is a figure. So here you have got the phase shift which is changing with energy E0 is the resonance energy E0 or ER here it is written as E0 because this figure is from Arno Bohm's book on quantum mechanics which is a very nice book which I recommend for this topic. So here ER is the resonance energy and as you see the phase shift changes rather rapidly between here it goes through pi by 2. So from here to here the net change is pi by 4 plus pi by 4. So there is a pi by 2 change in this region and there is another pi by 4 change as you go from well below the resonance and go to well above the resonance there is another pi by 4. So the net phase shift change is from delta to delta plus pi. So the net change in the phase shift is pi that is the angle through which the phase shift changes as you go across a resonance and very close to the resonance in the immediate vicinity of the resonance the phase shift changes by pi by 2 and that is where it changes rather rapidly. And how rapidly it will happen will depend on the width of the resonance. So gamma is the width so half gamma you reduce from E0 you get to the left of this to the lower energy side and an equal amount on the right of this. So this is the region where it changes rather rapidly. So how rapidly it changes that slope will depend on the actual width of the resonance. So this is the net change in the phase shift which goes from whatever value it has delta to delta plus pi. So as you go across a resonance the net change in the phase shift is pi very close to the resonance it goes through pi by 2. So let us consider scattering by a deep square well and I will consider angular momentum greater than 0 and when you have an angular momentum greater than 0 remember that when you separate the Schrodinger equation in the radial part on the angular part the radial equation will have a constant of separation which is coming from separating the radial part of the Schrodinger equation from the angular part of the Schrodinger equation you would have done that in the first course in quantum mechanics. The centrifugal barrier term l into l plus 1 by r square term comes out of this separation. Now this is a centrifugal barrier centrifugal in some sense it is a pseudo term it is not the result of a physical interaction like even in the hydrogen atom the physical interaction is 1 over r that is the Coulomb interaction. And when you write the Schrodinger equation the radial part of the Schrodinger equation you will have the net effective potential which is minus 1 over r which is the Coulomb attraction plus l into l plus 1 by r square which is the centrifugal term. So it is not the result of a real physical potential just the way the pseudo forces are not because of real physical interactions they are a result of the fact that we try to do dynamics in some accelerated frame of reference which is why the pseudo forces or the pseudo potentials shape up and here this term comes up because you have projected the three dimensional problem on a one dimension which is the r dimension you have separated out the spherical harmonics. So the two dimensions for theta and phi the two degrees of freedom that dynamics is separated out and now you are left with only a one dimensional Schrodinger equation namely the radial Schrodinger equation and in this radial Schrodinger equation you have a centrifugal term which is a pseudo potential kind of thing and this is repulsive. So this goes all the way to the to infinity as r tends to 0 and the net effective potential will then you know it will be but it starts diminishing as r increases as r goes to infinity it will of course go to 0 and then the net potential of the square well. If you set up the radial Schrodinger equation for the spherical square well potential then you will have a potential which goes like this square but then because of the centrifugal term you will have a barrier here. So this is the effective potential as you will see. Now what is happening is really interesting because if you have if you consider collision of a certain projectile by this potential and let us say that the energy of this projectile is e 1 it could be e 1 it could be e 2 it could be e 3 and whatever. So if you consider a particular energy like e 1 as you see in this figure then you can see that a particle with this energy would not have a chance of being bound unless the barrier was really high it needed to be high enough and if the barrier was infinite then of course you could have trapped it inside. So this would not be a bound state what about this state if the energy is e 2 then it has the possibility of being bound in the inner well but if it tunnels through this barrier region so from here to here is the barrier region and if it tunnels through then it could actually escape into the continuum with positive energy. So what is going to happen for a particle with energy e 2 it has two possibilities one is that it can be trapped in the inner well and the other is that it can exit into the open continuum and escape to R going all the way to infinity it can leak out and go all the way to the asymptotic region R tends to infinity. So these two possibilities both coexist and it goes back to the classic Young's double slit situation that you have two possibilities and then the amplitudes would interfere just the way they do in the Young's double slit experiment and then you have a resonance phenomenon because you have two possibilities which are both possible both probable and this is happening because of a particular shape of this potential. If L was 0 the potential would not have this shape and depending on the value of L L into L plus 1 by R square will change and the details of this shape will change. So here is a resonance phenomenon which is induced by the shape of the potential which is why these resonances or metastable states are called as shape resonances. So this is a metastable state it will have a certain lifetime because it has the possibility of being bound but it also has the possibility of being found into the continuum. Now another potential may not have such sharp boundaries like a square well potential you may have some sort of a this is like a harmonic oscillator kind of thing potential right and then you will have some discrete bound states and then if you have a state over here which is above E equal to 0 but not so much that it is well above the highest value of the potential. So this is another example of a shape resonance. So this is for more realistic potentials which are not like having sharp boundaries like a square well. So this is an example of a shape resonance. If you remember we talked about resonances earlier in S wave scattering and the S wave scattering those were coming because of the virtual bound states. So resonances can be because of very many different reasons. So the S wave resonances that we talked about in the context of the 11th's theorem and so on. So those were coming because of 0 energy resonances they were because of the virtual bound states or which we often called as a half bound states. But that was not because of a potential shape. So here is an example of a shape resonance. So this is called as a shape resonance specifically because it is determined by the shape of the potential what exactly is the shape of the potential. So that is what determines the shape resonance. You can have a resonance because of some other situation and that is typical in a many electron system because when you have a many electron system you can have some other kinds of resonances and these are if these come about because of certain correlations and why will there be such a correlation because a many electron system has got different electrons and all of them do not have the same binding energy. So you may have a process like in photo ionization or more specifically in auto ionization as we call it. You may have a photo absorption of a photon which is sufficient to knock out an outer electron but it does not have enough energy to knock out an inner electron. So what will happen? It will not be able to knock out an inner electron. It will knock out an outer electron however as you sweep the photon energy if you are doing spectroscopy and like at a cyclotron light source you carry out the measurements at different wavelengths or different energies. Then you may hit upon an energy which is sufficient not to knock out an electron from the inner level but to raise it to an excited bounce rate and at this energy you have two processes which are degenerate. One is a bound to bound excitation of an inner electron and a bound to continuum transition of an outer electron which could result in ionization. So now again you have two possibilities which are degenerate which can coexist which may happen at a particular energy which is the resonant energy because the bound to bound excitation is possible not at any arbitrary energy but only at specific energies because the bound state spectrum is discrete. So whenever you have such bound to bound transitions so you can have a bound to bound transition from a more tightly bound state but if this part of the discrete spectrum is embedded in the continuum of a less tightly bound electron then you will have a resonance because again you have two possibilities and once again you can think about it in terms of the Young's double slit that you will have interference between the two alternatives. So this is happening because of electron correlations because the two electrons the dynamics of one electron cannot be completely separated from the dynamics of the other electrons. So you have got the two electron correlation and as a result of this correlation which is not taken into account in an approximation like the Hartree-Fock because in the Hartree-Fock it is a frozen orbital approximation it pretends that whatever happens in one orbital has no consequence on any of the other orbitals which remain frozen but when you go beyond this approximation and think about the electron correlations then it becomes impossible to separate the dynamics of one electron from that of the other. So configuration interactions or electron correlations are then responsible for resonances as well. So you we first talked about the shape resonance which you can talk about in the framework of the independent particle model because here you are talking only about one single electron no second electron is involved over here. So you talk about the shape resonance in the framework of a single particle model of course electron correlations are there and they complicate this further but the fundamental process does not require correlation. The fundamental process is described in terms of the shape of the potential itself whereas the other resonance like the authorization resonance is essentially a correlation effect and you will then be talking about a many electron system with at least two electrons two is already many. So here this is an example of a many electron atomic system and let us take the example of neon which has got 10 electrons so 1s2 2s2 2p6 that is the usual configuration of the neon atom and you have the 1s is filled 2s is filled and the outer 2p you can have a bound to bound transition over here. So these are various bound state excitations possible from the 2p1 half which is more tightly bound than the 2p3 half. So if you have a photon energy which is more than the binding energy of the 2p3 half electron but less than the binding energy of the 2p1 half. So some binding and some photon energy which is greater than the binding energy of 2p3 half so that it can kick out an electron from the 2p3 half into the continuum. It cannot however kick out an electron from the 2p1 half state which is slightly more tightly bound. However it may be just enough to raise it to a bound excited state like the 4p3 half the 4d3 half of course you have to respect the dipole selection rules. So it can from 2p1 half you can go either to d3 half or to an s1 half. So you will have two possibilities over there and these are the two excitation bound to bound excitation channels which are possible for the bound to bound transition whereas the 2p3 half to continuum you can have various possibilities of getting into the continuum because from 2p3 half you can get into the d type continuum so d5 half or d3 half and you can of course go to the s half continuum. So these are the different relativistic channels and I do take into account relativistic splitting of the levels in this discussion because that is fundamental to the understanding of the resonances in this region. So you have the 2p1 half to nd3 half or n s half so n are discrete state quantum numbers of excited states and this these bound to bound transitions are embedded in this bound to continuum. So continuum state energies I represented by this epsilon bound state energies I represent by the principal quantum number n because the bound state energies in hydrogenic model go as 1 over n square. So this is a typical example of an auto ionization resonance. So you have two possibilities and there will be interference between the bound to bound states and the bound to continuum and these states are sometimes referred to as quasi stationary states or resonance states and these are these come into play because you have got one bound state spectrum which is embedded in the continuum with respect to another threshold. So this results the electron correlations cause these resonances which are sometimes referred to as the Butler-Fano resonances in collision dynamics. They are usually referred to as fresh back resonances or Fano fresh back resonances and so on or anatomic physics when you are working with outer state outer electron you typically refer to them as auto ionization resonances. Now the classic analysis of this is due to Fano and I really like to show a picture of this paper it is a classic paper it came out in 1961 okay physical review volume 124 number 6 1961 and this paper is one of the most cited paper in physics literature means you would think that okay the papers which are most cited are papers by some you know people like Schrodinger, Bonn, Bohr, Heisenberg and so on. But this paper is cited across all branches of physics atomic physics condensed matter physics solid state physics nuclear physics okay and this is a classic paper and it is I strongly recommend that you go through the original paper itself look at the number of citations in each year like in 2001 it was referred to 81 80 times then 2004 82 times 1997 84 times 2006 143 times the number of citations in FISRAV are well over 600 thousands of citations okay this is absolutely remarkable paper and something that you would love reading and when you go through this paper you will really understand why it is so important because it explains the fundamental process of correlations okay and how these correlations are to be analyzed in terms of configuration interactions and so on. So this is the classic paper and it results in resonances so the typical states that we talk about which are non-resonant states these are the stationary eigen states and if you have an isolated system then for an isolated system you have got the Schrodinger equation whose solution has got a space dependent factor and a time dependent factor which has got a uniquely defined sharp energy this energy E that you find in the solution of the Schrodinger equation is an extremely sharp level and if any electron gets excited to that state okay that state being sharp it will have infinite lifetime and it has no business to that an atom in that type of a state will have no business to decay and come down to a lower energy state okay but the reason it has got the sharp energy is because you have solved the Schrodinger equation for the atom pretending that there is nothing else in the universe that the whole universe is only this but then there is the rest of the universe and the coupling between the rest of the universe and this atom allows for the energy to be transferred from one to the other okay and this is just like having what we call as dissipation okay what we call as friction the friction when you rub something on a surface okay you say that heat is lost or energy is lost now energy cannot really get lost where does it go it gets transformed okay it is not really lost but then it gets lost from our bookkeeping because when we set up our equation of motion we have not taken into account the degrees of freedom coming from the surface interactions so these are the unspecified degrees of freedom and whenever you have these unspecified degrees of freedom you have the possibility of the system decaying from what you would otherwise expect to be a stationary state so this is not a stationary state it becomes a metastable state because there are these additional degrees of freedom so that the energy can escape to that so you have when you when you want to rewrite the energy of such a system taking into account the effect of the environment but not the details of the environment you can do so by writing these energies as complex numbers so that they will have a certain width and a certain lifetime so the lifetime comes from that not from the solution of the Schrodinger equation for an isolated system the Schrodinger equation for an isolated system will always give you for bound states sharp energy levels which will have infinite lifetimes okay so now you will have these quasi-discrete states because of the presence of the environment it okay so these are quasi stationary states or resonances or quasi-discrete states they are also called as quasi continuum because they are neither completely bound nor completely into the continuum okay so you refer to them sometimes as quasi-discrete or quasi stationary quasi continuum and all of these terms are used to convey one or the other meaning like an a thesaurus you have so many different terms which describe different connotations different meanings of the term so you have these different possibilities but you are all always talking about the same essential physics so there is a possibility of disintegration also because you have one possibility the continuum channel is degenerate with the bound to bound discrete excitation channel and the continuum channel will leave you with fragments of the system which recede which go away from each other into the asymptotic regions far away from each other infinitely far as well so obviously you will now run into resonant weights time delays lifetimes and so on right so these are the things that you will now have to be concerned with so the lifetime so this is like the energy time uncertainty which you are all aware is not the same as the position momentum uncertainty okay it is coming because of this coupling with the rest of the universe okay there is no operator for time in quantum mechanics so when you write the uncertainty principle for position and momentum there is an operator for position and for momentum but when you write the uncertainty relation for energy and time which are also canonically conjugate you do not write an operator for time because it is always treated only as a parameter in quantum mechanics so you have this weight which is given by a relation which is like the uncertainty principle it is in fact often referred to as the uncertainty principle but it fundamentally it is slightly different from the QP uncertainty because of this real difference because of the fact that time does not have an operator it is only a parameter so we will continue to refer to it as uncertainty principle because it has the same structure and the relationship is again between canonically conjugate variables and the description of this process is then possible in terms of the collision physics the scattering equations that we have set up the phase shifts and so on so we will discuss it in terms of the scattering phase shift so I mentioned earlier that utter resonance the phase shift changes rapidly through pi by 2 and from well below the resonance to well above the resonance the net change in the phase shift is through pi okay so across the resonance you will find that the phase shift changes through pi close to the resonance it will change rapidly through pi by 2 how rapidly it will change through pi by 2 will depend on the width of the resonance and some other details of the resonance profile if you look at the derivative of the phase shift with respect to energy okay so this is the energy derivative of the phase shift this energy derivative of the phase shift if you see from this profile so this is d delta by de which corresponds to this variation of delta with e okay notice that the phase shift changes most rapidly at the resonance and here the rate of change of delta with e becomes 0 okay it it it it is it becomes flat so so this is the rate of change of phase shift with energy the only thing you have to remember is that the change in phase shift which is a change through pi across the resonance we have plotted this figure as if we started out from 0 and then get to pi when we go through the resonance however the phase shift at the onset of this resonance may not be necessarily 0 it may already have some value which is coming from the background scattering okay and then over and above that background it will then go through a change in pi pi by 2 most rapidly at the resonance but it may have so the 0 of the phase shift for a detailed discussion will be offset so that is something that you must remember so here is an example of the phase shift here so here in each case if there are various figures here and this is also from Arnaud Bohm's book so here you have the phase shift changing through pi in all of these five figures okay but the starting phase shift is 0 only for the first curve for the second it is somewhat different for the third again it is somewhat different for the fourth it is different and for the fifth that is different so depending on what value it had from the background scattering okay because the collision process we have factored into two processes one is a hard sphere component okay so delta the net phase shift is the sum of two angles xi and rho xi is coming from the hard sphere and there will already be some phase shift because of that and over and above that because of the dynamics of the collision process you may have an additional phase shift go which goes through a change in phi and look at this that if you have a starting value which is different then if you plot the cross section which is like sine square delta for that particular partial wave of course these figures will be different for every different value of the L quantum number then the scattering cross section will have very different kind of profiles and you will see at a resonance when the scattering cross section goes all the way to the top or it goes up and then decreases comes back or it could go to 0 as you see over here okay so all these are possibilities so resonance does not necessarily mean that the scattering cross section will go to the top and go through the roof it can also go through 0 and this will not surprise us because we already know that in a Young's double slit experiment you have got bright fringes you also have dark fringes okay so that depends on how the phase and the amplitudes you know combine to generate the superposition. So we will take the example of this square well potential and we will discuss our phenomenology in the context of a very simple potential the actual potentials in that we have to deal with are much more complex but this is the easiest example so let us take up this example and here you now have this resonance width so this resonance width you have the resonance energy and depending on the width you go down half the width to the lower energy side and half the width to the upper above this energy and you define this angle as tan inverse of gamma which is the half width gamma by 2 over this E r minus E E E r is the resonance energy so you define this delta this is with a superscript r so this is what the other phase shift will be in the resonance region okay so your net scattering phase shift is the sum of two parts xi and rho which we saw earlier when you are at a resonance this will correspond to the resonance phase shift which is why to emphasize the fact that it is a resonance part so rho is not always a resonance because resonance will take place only is specific conditions for resonance are satisfied it can also be so both xi and rho in general are slowly varying functions of energy but at a resonance rho becomes such that it changes rapidly and that is when I refer to it as a delta with a superscript r to remind me that that is a special case of the resonance feature of the other dynamical phase shift which is rho this mouse does not work okay so outside the resonance region outside the resonance region the scattering phase shift is dominated by the hard sphere component okay so what relations do we get at the resonance if you go to this energy which is half width below the resonance the tangent of delta which is defined by this relation becomes plus 1 as you can see clearly right and if you go just as much above the resonance energy as below like go er plus delta then the tangent of delta in this case becomes minus 1 okay so the tan delta becomes plus 1 at this point and it becomes minus 1 at this point and this is what happens to the cross section itself okay for this type of behavior if it starts out over here okay so this is a typical resonance profile and you have a resonance width so the net change in delta will be through an angle which is pi what happens at the resonance energy itself delta this resonance phase shift will be pi by 2 okay it is here over here this resonance phase shift is exactly pi by 2 okay so it changes through pi by 2 from here to here at the resonance this what is on the vertical axis has got a value which is pi by 2 okay the net change from well below to well above is through pi so now you keep track of the parameters as we have defined them and essentially the change in the net phase shift will be from some value to another value which is pi above it okay so that is the kind of thing that happens across a resonance okay now at the resonance this is the tangent of the second part of the phase shift other than the heart sphere component right so see what happens as gamma L becomes equal to RL the tangent blows up okay the tangent of low goes it shoots up to infinity and the angle rho itself will be like pi by 2 or 3 pi by 2 or 5 pi by 2 and so on so that is the kind of angle or dependence that you get okay so this is the net phase shift which we have now written as a sum of these two parts these functions the heart sphere component and these functions R and S these usually change with energy but only slowly okay they are not nothing dramatic is happening to them what is happening at the resonance is that this gamma this changes rather dramatically and this changes dramatically as a result of which rho changes dramatically as a result of which the net phase changes dramatically and the net phase ends up going through a change in pi okay so the origins are in the details of the dynamics of the collision process okay because gamma is the logarithmic derivative of the of the radial solutions. So now we have separated the phase shifts into a heart sphere part and another part which depends on the dynamics of the collision process which could become resonant when conditions for resonance are satisfied and this solution will go into the scattering solution so this is the solution corresponding to the outgoing wave boundary condition you have got the scattering amplitude and the scattering amplitude is written in terms of what we call as a partial wave amplitude right so what comes in this beautiful bracket is what we refer to as a partial wave amplitude a is a partial wave amplitude f is the scattering amplitude. So what we are going to ask is can we separate out the heart sphere component from the partial wave amplitude that a itself if we write it as a sum of two parts one of which is coming from the heart sphere which causes the phase shift xi which is the heart sphere phase shift and the other part which will correspond to the dynamics of the collision process. So we try to separate the heart sphere component from the partial wave amplitude and indeed it can be done by looking at these complex numbers so this is not particularly difficult. So you write this phase shift rho which is now the second factor this is the second factor rho this is the heart sphere part the remaining part the dynamics of the collision is contained in this factor and the scattering amplitude is can be obtained from here. So the partial wave amplitude we want to write as a sum of two parts so this is the usual expression for the partial wave amplitude right so this thing in the beautiful bracket this is the partial wave amplitude and using this relation you can write this partial wave amplitude you can just separate out that part using this complex numbers and it comes as a sum of these two parts. So this is the heart sphere part and this is coming from the dynamics of the collision process. So these are the two parts of the partial wave scattering amplitude so this second part depends on the details of the potential the first part is just the heart sphere part. So this the first part you can write in another form in which you find it in many books so this is straight forward way of rewriting it. So I would not comment on this but you will often find it in this form which is essentially an equivalent form of writing the partial wave amplitudes. So now we will consider the behavior of the phase shifts across the resonance and we will take the example of a strongly attractive well. So you have got a well which is strongly attractive which is why I show it by what looks like a deep well just to indicate that it is strongly attractive a strongly attractive the strength of course depends on the depth as well as the range of the potential but you are only to indicate that we have a strongly attractive well and we have used these relations earlier so you define the quantum numbers kappa square which is the sum of lambda 0 square and k square is the energy of the projectile it is actually h cross square k square by 2 m but essentially k square is a measure of the energy right. So you have a deep well and the reason I refer to this is a deep well because kappa a in this case is much larger this l into l plus 1 even when l is not equal to 0. So as you can see from this right hand side the value of this right hand side will increase with the value of l but we are dealing with such kappa with such depth so that kappa a is always greater than l into l plus 1. So we consider low energy scattering and in this we look at the parameters r and s these have different values for different partial waves and you can write these hankel functions in terms of the Bessel functions the Neumann functions and look at the low energy behavior because that is well known and from this you can get the low energy behavior of the hankel functions and get the low energy behavior of the terms r and s. So that is a straight forward analysis which I will not work out in details but you can see where it is coming from. So you have the low energy behavior of the Bessel function and the Neumann function in terms of which you can describe the low energy behavior of the hankel functions in terms of which you can describe the low energy behavior of the functions r and s and in terms of this you can analyze the phase shifts the phase shifts rho. So you can get these relations that when you are dealing with low energy so that k is much less than 1. So this is the low energy domain and you can get the values for s when l is 0 and when l is not 0 when l is greater than 0 and you get these values from this by taking the low energy behavior and now you can examine what will be your gamma because gamma is 1 which is going to be a controlling factor. So you find when you put all of these values of r and s in this relation gamma which is nothing but this kappa j prime over j so this turns out to be given by this k cotangent function minus 1 over a. Now it is this cotangent function you know that the tangent function and the cotangent function they are very sensitive functions and there are regions where they just shoot up right and any small change in the angle will change their value in such a huge manner that you will have dramatic results. So that is what you will expect because the key feature in gamma is a cotangent function. So if you look at the cotangent function so this is the cotangent function and you see that when you are close to these points like if the angle is 0 or pi or 2 pi and so on any small change in the value of the angle changes the value of the cotangent dramatically and it will change the value of gamma dramatically which is why you have such spectacular changes when there are very small changes in the energy. So when you go across the resonance the change in the energy independent parameter may be very small but the effects on the phase shifts and on scattering cross sections are huge. So here look at it over here that whenever theta the angle theta is in the neighborhood of n pi whenever it is in the neighborhood of n pi there are huge changes in the value of the cotangent function and because of this gamma changes in a big way and because this gamma changes rho changes and then the net phase shift delta changes because delta is nothing but xi plus rho. So the xi part is not changing very much but the rho part changes in a very spectacular manner. So if you look at the asymptotes of this function they occur whenever this argument of the cotangent function what is the argument this angle is kappa a minus l pi by 2. So whenever this angle kappa a minus l pi by 2 is equal to n pi that is when you have the asymptotes of the cotangent function. So kappa a when this condition is satisfied you bring this l pi by 2 to the right. So kappa a will be n pi plus l pi by 2 that is what we get and if you look at neighboring asymptotes the adjacent asymptotes then for one kappa a will be n pi plus l pi by 2 for the next one which I indicate by a subscript n plus one instead of n here I have n plus 1 pi plus l pi by 2. So these are the values of kappa this is how kappa would change as you go from one asymptote to the next asymptote. So what is the net change in kappa a the net change in kappa a this is this delta is not the phase shift it only represents the change. So change in kappa a is pi or change in kappa itself is pi over a. So now you have kappa square equal to lambda 0 square plus k square. So from this we can take the derivative so you get 2 kappa delta kappa from the left and this one is just the depth of the potential so that does not change and then you have the 2 m over h cross square d because e is h cross square k square over 2 m. So here you get kappa delta kappa the 2 will cancel so kappa delta kappa will be m d over h cross square but now you have a delta kappa from here but you also have a delta kappa from here. So you can put the 2 together and what do you get you get that pi over a which is this which is equal to delta kappa but this delta kappa is equal to m d over kappa h cross square. So pi over a becomes m d over kappa h cross square. What is d? d is the energy difference between 2 adjacent resonances. So the 2 adjacent resonances will occur at this energy which will depend which we have now found to be given by pi h cross square over m a and it will then be related to the size of the potential lambda 0 is a root of the potential depth in this case. The potential depth itself is lambda 0 square so it is proportional to the root of the depth of the potential. So this is what you get for the separation between adjacent resonances and we will now be interested in examining how the phase shift changes at the resonances. So this expression we have obtained in one of our earlier classes so I will use it directly and in this notice that there is there is this gamma which we have obtained just few slides prior to this and now you can write this a gamma this is a gamma so we had obtained gamma so you multiply it by a you get a gamma and you can put this a gamma in this expression here and you get tan delta to be given by this relation you have neglected certain terms here. So this is your expression for tan delta this phase shift is has been written as a sum of these 2 parts remember that this k a to the power 12 plus 1 divided by d plus d minus this factor was nothing but the tangent of the Hartz sphere scattering phase shift. We have done this earlier this is the result from slide number 56 and we have obtained this result earlier so this term is nothing but the Hartz sphere component this term over here this term over here is exactly the same over here with the difference that there is a minus sign here so you have to be careful about it. So take care of this minus sign and you can write this phase shift so instead of this you can write this tan of xi because it is the same except for the minus sign and that minus sign you accommodate in the second term by making this l as a minus l instead of this minus a kappa cotangent function I have got a plus a kappa cotangent function and instead of this minus 1 I have got a plus 1. So I have just read the minus sign over here. So you have this net expression for the phase shift in which you have one part coming from the Hartz sphere impenetrable sphere and the remaining part is coming from the dynamics of the actual potential and that is a part which is of interest in the resonance condition and you can see where the resonance will take place the resonance will take place when this denominator will become small. The cotangent will take all kinds of values and when the cotangent takes such a value that a kappa cotangent theta where theta is kappa a minus l pi by 2 becomes equal to minus of l that is when you will hit a resonance. So we will discuss those details in the next class. If there is any question for today I will be happy to take otherwise we will pick up the discussion from this point in the next class.