Lec 2 MIT 5.74 Introductory Quantum Mechanics II

means you had the states switched yes that cut you you meant the composition of states does that mean the the basis states that go to form the the the the states that make up those curves switch is that what you're saying yeah um I mean you can so let's uh when you have a weekly avoided Crossing you get this kind of a a behavior and so up in here this would be a uh an electronic state has an electronic configuration which is the same as the or similar to the electronic configuration down here okay and so uh what happens is you have a mixture you have say configuration number one configuration number two and near here all of a sudden uh you go from almost pure configuration 2 to almost pure configuration one and right here you have a 50/50 mixture and so the electronic character is hardly changing at all until right here where it changes very fast and that that leads to gross violations of the Assumption uh that the wave function is independent or approximately independent of internuclear distance and as a result there are large coupling that uh um are outside the zero order model and you have to put them in in order to get the truth and so when you have really really weak uh configuration interactions really really weak electrostatic perturbation then you get this cusp if it's really strong you get a kind of behavior sort of like this and uh you hardly know that there was of an avoided Crossing because the two halves of the avoided Crossing are so far apart and because uh there is so little anomalous curvature you don't really know that that this is different from a simple picture of what you expect to go on okay and so and that also tells you that probably the ADBA curves are the good way to think about the problem and another way of knowing that is if you have something like this half the distance of closest approach is the electrostatic Matrix element which is enormous and you don't want to have something enormous off diagonal in your model it'll still work but the model won't look like or what's on the diagonal of the model won't look like what you've observe so this is I mean it's really important to distinguish between the model which puts and in fact this is what I left off I didn't uh have a chance to finish properly at the end of last lecture and uh uh so we have a model where the Hiltonia might consist of three terms we'll call them A and C but one might be electrostatic one might be spin orbit one might be root and uh you have a choice of what you put into the zero order picture you could say for one case you have a + b is a zero order picture and in another case you might have a + C it implies that B and C are incompatible and so in your zero order picture um you're out of luck you don't uh okay uh so uh we'll make the copy and we but what I'm talking about right now isn't in anybody's lecture Nots um what why don't I but before we go on why don't I circulate this uh list if you want a PDF copy of the typed lecture notes after they're proof read and so on put your name and email address here and Peter will send them out so the first lecture notes are nearly ready and the SE one second ones will probably be ready tomorrow okay so B and C are operators which are incompatible in other words you can't have states which are I States or of both B and C or you can't find a representation in which B and C are both diagonal you can diagonalize a and b simultaneously and A and C simultaneously but not B and C these are two mutually incompatible terms in your picture and you always make a choice there are always going to be mutually incompatible terms in your model for what's going on and you have to choose which one is the one that is better to put into the zero order picture so you'll capture most of the important structure that you're going to see in the experiment and then you put the other one in as a perturbation and you then uh use the I states of the part of the problem you've chosen to make H zero to evaluate the effects of the missing part and this is basically that's all spectroscopy is in fact I would imagine that's pretty much all uh any representation of a complex situation is that you you say this is the part that that is uh that captures most of the problem and that I can describe it in a convenient way and then I put in the other part as uh some kind of a perturbation term which I also treat accurately and honestly and completely but only as a modification of what is the fundamental Simplicity of this particular particular situation and let me uh uh be a little more specific in the hamiltonian for a diatomic molecule we have a contribution from the spin orbit operator the spin orbit constant times lzs this gives rise to the spin orbit splitting of a multipet state and we also have a term from the rotational hamiltonian the rotational constant times uh J + s- say minus s+ where this is called the spin uncoupling term this is off diagonal between states of different values of the projection quantum number JZ and the projection quantum number for SZ this is called Omega and this is called Sigma so we could choose a basis set where this is diagonal and as a result this goes into h0 and this is not diagonal and this term we we call the perturbation term and it leads to small modifications of a characteristic pattern which is easy to recognize and that called hun case a or we can put this in and this forces us to a situation where the good quantum number is n which is J - s and then the rotational levels go as B * NN + 1 and there's nothing from the rotational hamiltonian that's off diagonal but this is off diag and that's called H's case b now let me also take something from the last page of the notes from last time that we didn't get to so let's say we have two hamiltonians hamiltonian number one which has the form Delta 2 0 0us Delta 2 in other words splitting of the states on the diagonal of Delta so there's some term in the hamiltonian that lifts a degeneracy that creates a pattern and there's going to be another term in the hamiltonian that tries to destroy that pattern now for a 2X two it's easy to do all the algebra but generally the the problems we deal with are a little bit bigger than 2x two but not much bigger but the pattern is much harder to see and the amount of crank turning to to do the algebra is much more severe so this the other hilon uh is 0 0 BB this is an off diagonal hamiltonian with no difference in energy on the diagonal so our full hamiltonian is the sum of these two now we can use a basis set where this hamiltonian stays diagonal but then there's going to be this thing here and we either do corrections by second order perturbation Theory which would be correct the energies by v^2 over Delta provided V over Delta absolute value is less than one that's the the uh convergence Criterion for ation Theory when V over Delta is not less than one then we actually have to diagonalize this Matrix now this is an easy Matrix to diagonalize because the off diagonal the diagonal elements have the same energy I can write down the transformation to diagonalize that without thinking so what I'm going to do is go to the basis set or go to the igen vectors of this okay and those are 1 > 2 * 1 + or - 2 where these are the I vectors of this okay is for the people who only had 561 or who haven't had 561 this is a reach but I want you to get this for my students this is not a reach but this is really a crucial Point okay so this will diagonalize this honia and how do we know that well let's calculate the Matrix elements so okay so this is the symbol let's take the Diagon the the plus with the Plus or the minus with the minus so these are actually two elements at once and this is plus with the minus or minus with the plus and uh so we're going to have to wave function twice so we'll get 1 2 SAR so we'll get a half and then uh we'll have h211 plus H2 22 plus or minus h212 plus orus H2 21 okay so in H2 these two things are zero and these two things are both V and so we get plus or minus V and when we go through so we're just putting this in and calculating the Matrix elements of these things now I can do this sort of almost in my head but this is going to require a little bit of study from people who haven't done it before okay but I want to write it out so that you can convince yourself that you can derive that it would have been easier if I just wrote an extra line on the black board okay uh for the second one we have a half time h11 uh for Hamilton number two- H2 2 2 minus or plus H1 2 plus or minus 2 one and these These are zero but this is V and this is V and so whichever choice of sign we choose we get zero and so H2 transform give it that little symbol looks like this V 0 0 - V we go through the same thing for H1 and I'm just going to write that down uh for H1 when we transform H1 we get 0 Delta 2 Delta zero doesn't it look interesting this looks almost like the old H2 we have the same number off diagonal nothing on diagonal and this looks like the old H1 we have the same number but with opposite sign of the diagonal so if we choose to choose one basis set we have one term on the diagonal creating a difference in energy between the states and the other term off the ago trying to spoil that picture if we choose the opposite basis the roles of the two terms the good term and the bad term switch and good becomes bad and bad becomes good and this is everywhere what is happening and sometimes uh in order to understand the mechanism of what's going on it's necessary to make one of these Transformations and the reason for some off diagonal interaction between states that leads to some Dynamics or some complexity in the Spectrum is much clearer in one representation as opposed to the other and uh one example of that which we might get to today and might not get to it all is rotational Auto ionization I'll get to that I'll talk about perturbations now okay so this is what was left from the last lecture and it's probably the most important point from the last lecture we have always mutually incompatible terms in the hamiltonian and we choose one to go into h0 and the rest to put uh off diagonal and treat as perturbations every choice when you do the problem completely including all the off elements gives an equally good representation of reality but the different approaches when you ignore the alagon give very different correspondence to reality an appropriate representation gives a recognizable uh description of reality where there are small deviations okay The crucial actor in our treatment of molecules was the Bor Oppenheimer approximation so I keep dividing up the Hamilton slightly different we have the electronic plus vibrational we have spin orb and rotation the reason I put these together is that when we do the born Oppenheimer approximation we generate the potential curves and once we have the potential curves we solve the vibrational problem and uh whether we make a diabetic or adiabetic choice we get different potential curves so the vibrational problem is different but it's the same crank you turn you have a potential energy curve and you have the nuclear kinetic energy you solve the nuclear shorting equation you get IG functions and I Val okay the most for diatomic molecules the most convenient from the point of view of writing down Matrix elements and representing the problem the most convenient basis set is generally the Huns case a basis set and so case a basis functions look like this we have the electronic uh a symbol that says which electronic statement come with an i and then we have L which is never really good but we often include it as a uh label Lambda s Sigma and then Omega k m so these are the quantum numbers and these are our basis functions and in order to make any progress with diatomic molecules we need to know how to evaluate Matrix elements of all operators in the hamiltonian in this basis so for angular momentum Matrix elements you want to look at Pages 72 to 81 for my book which I handed out until I ran out of copies uh some of you haven't it's it's in that handout uh there's three pages that are with nice pictures which I'll get to and then there is the Machinery okay so let us say that a is an angular momentum and some operator some angular momentum and the nice thing about angular momenta is that once you know what the Matrix elements for one angular momentum are you know them all and so we have the angular momentum Square operating on a basis function which is expressed with two quantum numbers the quantum number associated with the total angular momentum and with a projection of that angular momentum projection on what we call the the Symmetry Axis or the z-axis okay we're two free to choose but in diatomic molecules you would be kind of stupid to choose anything but the internuclear axis of Z because that's an axis of cylindri iCal symmetry but it turns out that for ridberg States sometimes the quantization axis is not the internuclear axis but it's the axis about which the molecule rotates and that leads to some Strang but so you have a magnitude and a projection and so you get a a +1 a Al we have the space fixed the the laborator sorry the body fixed zis projection and that's Alpha a Al then we have the raising and lowering operators which are a X plus orus I a y and a +us on a alpha gives a a + 1us alpha alpha Plus or - 1 square a plus- 1 and that's all that's all you need this is a you know a small amount to learn in order to be able to address practically any problem in diatomic molecule spectroscopy and the reason case a is nice is you have three important angular momentum and each of them have their projection specified so so that means it is completely transparent to write Matrix elements of a s a z and A+ minus there isn't anything else so you don't have to use CL Cordon coefficients or 3j coefficients if you're in case a so even for situations where case a is inappropriate many people prefer to use Cas a because there it's easy for them to set up the machine that represents their spect now there's no big deal using 3j coefficients to deal with angular momenta coupling and uncoupling and if you're going to be a professional you want to be able to work in different basis sets and to transform between them using the uh the vector coupling or 3j coefficients um and you can get a computer routine that will calculate any of those you ever need and so the only difficulty is that 3j coefficients are expressed angular momentum transformation coefficients are expressed or are labeled with six quantum numbers and so somehow you have to think about six quantum numbers and after a while it becomes not so hard but this this is a more linear approach in some sense okay so I said enough about that uh if you're going to start out doing uh datomic molecules I strongly recommend case a yes you remind us I can't remember is there any obvious way to to look at a datomic molecule and guess whether it's going to be case A or B is it has something to do with the size type of you have to look at the versus B uh J okay so all you have to know is this [Music] orbit the rotational Conant that's right and so a is small for the beginning of the periodic table it gets larger and larger and uh B uh is uh large for hydrogen and large for molecules that have one hydrogen and sort of intermediate for molecules that have two second row atoms like Co and uh for heavier atoms B is very small and so as you go down the periodic table a heavier heavier molecule you tend to go towards case a but if Lambda is zero in other words Sigma States you're always in KP but it's not a big deal okay now I said it was easy but there's one little patch if you want to do this without thinking you have to say oh uh Bob told me that if we have an angular momentum that controls the RO the rotation of the body fixed coordinate system relative to the laboratory coordinate system then I have to make a small patch and it turns out that j n and R J is the total angular momentum n is J minus s the total angular momentum exclusive of Spin and R is the nuclear rotation then J plus lowers Omega instead of raising it n plus lowers r+ well r+ can't really do anything because there is no projection of r on the inter nuclear axis because R is we have Atomic molecule and R by definition is perpendicular so but these angular moment have some special property and you can you can understand that when you look carefully at the transformation that takes you from body to laboratory Direction cosign and when you work when you develop more facility with the the true classical mechanics and one mechanics of the system okay but right now I just say this is a patch and basically you are now empowered to do all calculations on datomic molecules at least as far as the angular meor are concerned okay now if you want to deal with the nuclear motion that's another topic which I will address okay so now I make a little [Music] table of all the angular momenta you're likely to encounter well a lot of them many is so we have the operator the quantum number and the [Applause] selction so j s j z J plusus I'm not going to be able to do this all on one board but you'll get the idea so operators are generally represented as a bold face oops plus um J has two projection quantum numbers one in the laboratory and one in the body and so I've chosen to use the Su superscript for body and subscript for laboratory you encounter that less often uh and so I'm not even going to talk about it so we have the quantum number is J and for JJ for J squ you get JJ + one and quantum number are generally written as I tell it the selection rule for j s is Delta = 0 Delta Omega = 0 the quantum number for JZ is Omega Delta J = delega there's no quantum number for this operator because it's an off diagonal operator and the selection R is Delta j = 0 Delta Omega = minus or + one similarly for s² s z and S + s s +1 Sigma Delta S = 0 Delta Sigma = 0 Delta Sigma S = 0 Delta Sigma = 0 Delta S = 0 Delta Sigma and we can go on for a long time now we come to l or l squ because a molecule isn't spherical L is usually not defined but we can still Define out Z and we can still use l+ minus so we don't have a quantum number here although we'd like to write LL + one and we we don't have a a qualum number for a selection rule for L because we don't know what L is but we still have Delta Lambda equals z and here we still have Delta Lambda equal Z and here we have Delta Lambda plus orus one there are other quantum numbers like or operators like n which is J - s and it turns out that n z has the quantum number Lambda because NZ is identical to LZ as far as the part that can have uh a projection because NZ uh um we I mean JZ is LZ plus SZ plus RZ RZ the projection of the nuclear rotation on the inter nuclear axis is has to be zero and so uh JZ minus uh JZ minus SZ is LZ so LZ is NZ uh okay now we get to more esoteric situations uh but I want to talk about them suppose you have a molecule uh which has sort of a a a closed shell or nearly close shell core an electron far away from it and so this is an ion and this is called a redberg state when the electron is far away from the ion it thinks it's on an atom and we get L back so but we get not Big L but little L the L for this rer electron and so we start having more quantum numbers we have the quantum numbers associated with all of the angular momentum of the ion and you have the quantum numbers associated with the rber so instead of having uh just some uh L S and J as our quantum numbers we start having six quantum numbers we have L plus S Plus and j+ and little L little s uh and Little J or something like that so we still can couple them together we can still use case a methods uh and so we we can have j+ now this is not the raising or lowering operator this is J for the ion we use plus and you can tell the difference because when you have a bold operator a bold symbol it's an operator and when you have an italic symbol uh it's a quantum number and so uh and when there's a problem uh when you have an operator and you want raing and lowering operators a bit and you put okay and so uh we would uh we would the the quantum numbers associated with uh the projection okay so suppose we have the J + z that would be called Omega plus the quantum number okay this is more detailed than you need okay so we have we just put a Plus on the quantum numbers for the ion and since this is a one electron quantity we use lowercase letters Little J little L little s little little Omega lowercase Omega lowercase Lambda lowercase Sigma for the quantum numbers of the electron so there isn't any ambiguity there's lots of angular momenta and there's lots of coupling cases remember when we talked about the coupling cases last time I said we had three turns electronic spit orbit and rotation and we can arras them in six orders of relative size and that leads to the five known Hood coupling cases and one other but now if we have more quantum numbers you can imagine how many coupling cases there are but it's no big deal because the same techniques you use to evaluate the Matrix elements and rearrange terms of the Hiltonia there's nothing different okay so you are empowered to deal with angular and diatomic molecules even if we have six rather than three important angular moment so what's what's left well there's the vibrational equ and let's talk about them and I'm going to present a very very important approximation that enables us to uh simplify our work dealing with vibrational matrixes okay so the vibrational wave function we have that should be a Kai we have a vibrational wave function for the I electronic State and then it's got a vibrational qu is the function of r in direct notation that object is the same thing as this okay uh we tend to like to think of vibrational wave functions as functions and angular momentum wave functions is vectors because we don't what I mean this is a simple one-dimensional function and we're going to be calculating integrals over it there aren't nice uh angular momentum like algebra for vibrational weight functions whereas angular momentum wave functions are treated by an algebra where basically all the Matrix elements are directly evaluated and you never look at the differential operators that correspond to the angular momentum nor the forms of the wave functions so it is tempting to look at these sorts of things rather than this kind of a symbol for vibration now we have a wave function which is a product of an electronic vibrational and a rotational part so in order to calculate Matrix elements we're always going to be evaluating uh a product of three factors a rotational Matrix element a vibrational Matrix element and electronic Matrix element so the electronic part we deal with first because uh we get rid of it almost immediately so we have some electronic operator and we have two different electronic States and according to the Bourn Oppenheimer approximation these functions are parametric functions of internuclear distance so that means that we're going to have this Matrix element is going to be a function an implicit function of r inter so the borop says well we do the electronic integral it will be a number but this number will probably weakly depend on R now if this operator has an explicit dependence on r that also appear here now one operator that has an explicit dependence on R is the rotational constant HR S 2 u r s now that's not an electronic operator but so we're often going to have things that either implicitly or explicitly depend on intern includ distance this is a problem because when we have an electronic property that depends on our and we want to evaluate Matrix elements now I am uh writing the vibrational wave functions as as bro and KS here so this is a vibrational Matrix once we've done the electronic integral now we have to do the vibration interval if this is a function of our we got B problems we like to ignore the dependence on R so how bad is this how bad a problem is this well one can always expand this as a power Series in R and then evaluate all the necessary integrals so that would mean we have have uh things like just the vibrational overlap we'd have v r VJ we'd have VI R 2 VJ we'd have to calculate lots and lots of vibrational Matrix elements and this would be inconvenient and it would also be costly in terms of our understanding of the problem and so we'd like to make that go away so it turns out that we can always or almost always replace this kind of an integral with VI VJ a simple vibrational overlap times o i j of r v i VJ r that's called the r centroid the r centroid is defined as it's this quantity The Matrix element of r divided by the overlapping that's perfectly reasonable it it is the way you would Define the average value of R so the r centroid this is the r centroid the r centroid approximation says replace this integral of the function of r by the overlap integral the only vibrational integral you're going to have to calculate we not the only one times the r centroid this o of the r centroid instead of O of R that's an approximation it's a very good approximation and it's based on the assumption that r r v i VJ is equal to r n r e DJ over r nus1 e DJ this is the r centroid approximation the r centroid is this is given by any value of any power of n this ratio n n minus one now why should that be true it's astonishingly true how can we expect vibrational integrals involving different powers of R to have this property this gives us a hint of something that's going to be really important the integrals accumulate all always in a very narrow range of art that's called the stationary phase approximation and we'll get to that but that leads to really important simplifications because often one can know explicitly what region narrow region of internuclear a distance the integral will accumulate it for example if we have a problem like this where we have two potential curves and they cross and we want to calculate the vibration overlap between these two levels at about the same energy they integral the vibrational overlap and any integral involving any power of R will accumulate in a narrow region at the internuclear distance where the two curves cross and so for all vibrational Matrix elements there is hiding in here this stationary phase situation this just means that when the two wave functions have the same nuclear kinetic energy they're oscillating at the same spatial frequency they have the same momentum and the momentum gives you the de Bry wavelength to the BR wavelength Lambda is uh H over e and so if they have the same kinetic energy they have the same momentum they have the same wavelength that means that at this point they're oscillating at the same frequency and so the integrant uh which uh is the product of two wave functions that integrant oscillates very fast except near this point and that's why the r centr approximation works and we'll return to this when we talk about the semic classical approach to uh everything okay but the rcent approximation is very convenient because in order to deal with um these kinds of Matrix elements one has to calculate only two integrals this one and the integral involving R the r centroid is this over that uh is the uh yes so there's only for every pair of VI and VJ instead of having to calculate Matrix elements of all powers of R when just calculates the overlap and the Matrix element of R to the one power and that leads to tremendous simplification of effort and it also allows us to describe things in a way that's much simpler the observables s the AR dependence of electronic quantities only in the vicinity of some special point which you know and so they're not terribly sensitive to curvature to strange variations and so one can ignore the r dependence of electronic quantities for most calculations this is very helpful and it would not be clear if you were simply doing numerical calculations because you would always do the numerical calculation correctly and you'd be doing you know you turn the crank and youd get something but you wouldn't know that everything interesting is happening in a particular place and the nature of your wave function at that internuclear distance is what determines what happens in the experiment and a different experiment might have curve cross at a different R and it might sample the away from a different place and give you something else okay yes so where would this would break down if you had say uh two peaks if your wave function had two maximum and they would that be a case of where it breaks down no it'll break down when you have something like this where there's no spatial overlap between the two vibrational WS an R will be in the middle it we'll try to be somewhere here and the the so it's in the Tails of the wave functions and so there's just no get a weird value for the elect that's going to be really high because it's really high on those curves and the wave functions aren't even there that's right so when you have when you're dividing by zero you don't expect to have very much of a purchase on whatever it is you're King so then the overlap integral is really small for a good reason right as opposed to because of an accidental cancellation then the r centroid approximation really fails I was just wondering if you had if the wave functions happen to have two Maxima and those happen to be if if they had two maximum basically essentially what you're calculating is the average value of r with the the that first with the r centroid right and so if you had two Maxima of the wave function the average value of R might turn out to be in between them that's okay because the wave function isn't zero except at nodes right and so uh what you're basically doing is you're measuring at this special Point what is the classical probability which is related to the classical velocity or one over the classical velocity which is related to the classical moment momentum so you can uh you can still have small vibrational overlaps and still have the AR centroid approximation work because it's basically telling you that it's at this place and the the two uh the two vibrational levels correspond to the particle moving really fast in that region but there's it's still classical you still get the results by using classical mechanics in account in a semiquantum mechanical way it's almost always true that you can use classical mechanics embedded in a quantum mechanical framework and get the answer and when you do that you get what you're told you're not supposed to be able to get from quantum which is uh where does something happen in quantum mechanics it says this is the probability of something happening but in classical mechanics you're watching X and P as a function of time and you can say something happened at this position and at this time and the particle was going this fast and so you know what the forces were and you know you know so everything starts to have interpretation when you can embed classical mechanics in quantum mechanics so quantum mechanics says you're not supposed to know this stuff but usually you can and whenever you have an approximation like this saying you know everything accumulates I mean the only way you can have the r centroid approximation work is if things accumulated at a specific r I think this is what I mean Dave's question kind of recap is if you have two different potentials yes and they both have all their you know you're high vibrational State all the probability of two sides of both yes you're calculating an over electronic integral between those two yes you're going to say the are centroids in the middle and there's nothing there because all the action happens at the sides of both but classically uh you know how much you you know what I mean there is nothing there there's a little bit there and you know exactly how to calculate that because in the middle region where things are going really fast it's a lot of oscillations and uh so there's not going to be but then you're saying that the electronic whatever electronic integral you're calculating is determined by that point when you think would be determined where by where all the vibrational amplitude is the turning points but you would be it's determined where the stationary phase is and it's determined by how much time the particle spends in the stationary phase region what does that mean well we know a velocity and we ask well what is the stationary phase how far can it go before the two wave functions get pile out of phase or accumulate a pi phase error so the integrand the integral will accumulate about the stationary face Point overall length which is determined by how far you can go before the two-way function Pi phase difference and that's related to the difference in slopes of the potentials and the the value that accumulates is the classical envelope which is one over the velocity and so you can basically evaluate this integral not by numerical integration but by classical mechanics you need to know the period of oscillation the velocity and the stationary phase point and how long is the stationary phase region and all of those you can get simply by looking at the uh um potential burs and the fact that the wave functions have big loes that don't overlap with each other it means that the the integral is going to be small but the value of that integral you can calculate and you can use very simple ideas to get these integrals to factor to accuracy or you can use a functions to get the integrals to whatever accuracy you want and to have all of the you know the dependence on energy and other stuff are you I still what if what if you have two stationary phases if you have two if you have two stationary pH points you're screw I'm not yeah I'm not saying that you have each weight function separately has a lob two stationary well then and then you end up with an average in the middle that's that's right that's exactly that's if you have two stationary face points then semi classical arguments will not help you because each one is going to generate a large contribution to the integral right and you don't know their relative phas and you end up I mean it seems like our average is in the middle even occurring both at the Ed when there are two stationary phase points if you calculate the r centroid for uh several different uh uh successive vibration it'll jump all over the place okay so there will be warnings okay so that's your warning is that you're so and you can tell you you have two potential curves um and you subtract the two and you look at the difference potential as a function R the stationary face point is where the difference potential has zero derivative and if there's two or three places where it has zero derivative you're warned and sometimes you can get away with it because there's only large amplitude at one of those uh if there's large amplitude at Several of them or small amplitude at all of them then you're going to have to worry about the relative phases and that's where quantum mechanics does stuff that classical mechanics can't touch I mean there isn't interference in classical mechanics you can still use classical mechanics to guide you to the one thing you have to get from the clchs but often there's no point once you go that part you might as well use the qu okay uh we still have a few minutes okay now we're on page five of my notes so the born up and armor approximation tells us how to begin and uh usually life is a lot simpler than you have any reason to expect that the Spectra are really regular that there are nice simple patterns and uh uh that there is no hint that bad things are happening or could happen and so there's two limits one is the limit where H I J the op diagonal matrix element between two states that might be near each other two zero order born alimer States or case a diabetic States or whatever so the Matrix element over the energy difference is small so this is the situation that leads to unexpected Simplicity undeserved Simplicity there are all sorts of off diagonal matrix lurking but none of them lead to observable effects or very important effects when this is true one can always correct uh the energies by idation so this is Delta e um so the energies are corrected by this this is small it usually varies with uh State smoothly and so you don't really see anything weird but then there's the other limit where this quantity is much much larger than one and that's usually because of an accidental degeneracy so there's nothing really in the hamiltonian that says this is going to happen it's just the levels which you've normally been ignoring interactions between accidentally fall near each other and whenever that happens you can't use Ordinary second order perturbation Theory you get local effects and these are called perations the this regular thing is messed up there is something that looks wrong except there's nothing wrong about it U and but it requires a special treatment because of this unusual energy denominator that brings these zero order States close together and forces you to deal with the true evil of nature okay there's there's many types of perturbations one is we have a bound potential curve and another bound potential curve and we have vibrational levels of this and vibrational levels of this and they're near each other and then if we look at the energy versus J * J + 1 we get something like that remember uh the energies versus J J + 1 will appear as straight lines with slope of B and if you have different rotational constants these curves will cross so at this place where we have states of the same J that are degenerate something bad is going to happen and now suppose Nature has said well I can to this level because like we have a singlet ground state this is a singlet state but this is a triplet State I can't exite to that this is called a Bright State this is called a dark State and so uh uh if I um well now there's some pictures that I handed out that uh um um this uh what is a peration here is a spectrum recorded on a photographic plate which nobody ever does it but it it uh makes it so clear you see this regular spacing of lines that's an unperturbed Spectrum that's a textbook Spectrum there's no Dynamics in it nothing interesting and that's what most people would like to see because they're risk ofers and then this is just the next vibrational level higher in that same electronic State and you can see there's something wrong it's not there's no regular spacing of lines there are lines that have been shifted around and there's extra lines there's more lines in here you see how many lines in this lower you see they they're not just there's many closely spaced things and that the reason for that is because uh you're not just seeing the bright state but you're seeing the dark St that because of this accidental degeneracy there are Level shifts and intensity borro you open a window onto what you're not supposed to see and some of us still get a thrill when we can look through a window we're not supposed to see through uh all right so uh we learn about other states uh we learned about the coupling Matrix element we learned about Dynamics okay so I'd like to say more about peration but there's another kind of perturbation so we might have a bound State crossed by a repulsive state so this leads to a b going to A+ B this leads to a Breaking of the bond pre dissociation so we have a vibrational level here and a Contin vibrational Continuum here and because there are coupling Matrix elements between the bound State and the Continuum what happens is if we look very carefully the bound state gets broadened might get shifted and since there's Continuum above it and below it it is not likely to get pushed one way or the other by more density of States above or below so that level shifts are not important but level broadening occurs and again on these pictures that I gave you it shows um the second picture is still another perturbation but it's a small one but this third picture um shows Something Happening Here is the absorption spectrum and here is the emission spectrum same band in the absorption it doesn't care if the excited state can't live to emit a photon but in the emission spectrum if the dissociation rate becomes large compared to the spontaneous emission rate you get no fluoresence so this is a predissociation and it's interesting if you look closely that there's a sudden drop in intensity and then it picks up again that the uh the dissociation rate is uh slowing down as you go away from the the first J that is hit and so you you get uh a large variation in these mixings so we can still describe Creed Association by calculating the electronic Matrix element times the rotational Matrix element times the vibrational Matrix element and the Complicated new thing is we have a bound vibrational State interacting with a Continuum vibrational state but there's still a stationary phase region and that's determining everything um and there is another kind of perturbation that leads to what we call automization so we have a process a going to a plus plus an electr this is harder to draw basically what happens is we have an electron bound to an ion and the ion can have electronic exitation spin orbit vibrational exitation rotational and somehow the energy in the ion Finds Its a way to get into the electron to transfer to the electron and when when it does that the electron is too much kinetic energy to remain bound to the ion it falls off and there are beautiful mechanisms that describe how this Auto ionization occurs and each of the processes uh electronic spin orbit vibrational rotational Auto ionization has a nice mechanism