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# Lecture 08. More on Vibrations and Approximation Techniques

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harmonic oscillator all perfectly evenly spaced all the way up as long as we have this potential that goes like that but of course we don't and we we only use the harmonic oscillator near the ground part of the potential where we expect that we can approximate the force constants as harmonic when the atoms get too close together the repulsion is very strong and so the inside part of the potential goes up much steeper than quadratic and the outside part starts to fade out because when the answer to far apart they don't attract each other at all and so they're in Real a molecule solicited diatonic there's a strict limits on what Delta are can be or what we called X 1 we wrote our function the way functions for the higher energy Oregon State of the harmonic oscillator turn out to just be some polynomial in Annex times our same Gallician functions and what the polynomial does is it introduces notes it's a common theme throughout that 1 you go to a higher energy you have more and more nodes in the wave function just like if you have a higher note you have a shorter wavelength on the guitar and these particular polynomials happening there called the Hermes polynomials they're well known in mathematics and in fact lots of polynomial functions sets of polynomials that have names come from solving some particular problems in some field of physics and chemistry and then whoever solved it gives their names to the polynomials for other people give give the name in honor of the person if we use this shorthand here but that I'm going to introduce just to kind a tidy up the formula so that they are too big to fit on the slide we introduce this variable y which is and will make a divided by age take the square root of that times effects was kind of a scaled distance than that allows us to to tidy up the formulas and and write them in a much more compact way so will introduce this variable y and then we'll have a look at what we here then In terms of this variable y are the 1st of 3 of 4 energy levels of the harmonic oscillator the ground state is the guards union the 1st excited state has next determine why multiplying by the GAO 1 why is 0 X is 0 as well and so in the middle of the oscillator we've put a node rather than having the function come up like this turtle we've put a node and so we get something like that and if we go to the next 1 we see that we have even powers and why so we now have won over the square is 2 times to y squared minus 1 that is an even function the function wise not function the first one was immune function just the go and it won't surprise you that as you go up the ladder of levels that you get you there and even function or an odd function it has to do with the parody which turns out to be an important concept in more advanced courses the parity changes depending whether the function changes signed from next minus X or has the same signed from next mine 6 and the the 3rd 1 is too wide to -minus 3 why that's of particular form they keep going and there is a convenient way to generate the we can apply something to on the last 1 to get the next 1 and you can mechanically generated all these things because just figuring them out from the get-go would be extremely tedious ah book has good plots of these functions so I won't make more plots here they they basically look exactly like you would think they have some wittily parts and then they die out but as you start introducing higher powers of why what happens is you start having things get big toward the edges were why gets big before it dies out either gets bigger negative effects on on function
or gets bigger positive on both sides of it's an even functions and that kind of corresponds with what we normally think of as the way an oscillator works namely that the probability if we take a snapshot of the oscillator is that we're going to be either fully compressed by because then we are moving much and so that the percentage of time that were there if we take a snapshot very likely they were going to be there or we can be fully stretched his we have to turn around there as well and so we have to stay there awhile to make a U-turn and come back what we don't expect to find is to find the thing in the middle a lot 4 an oscillating harmonic oscillator because in that case you're moving fast and so the percentage of time you would expect to be there or equivalently when you measure the position at some random time would be small I the let me just say that the Hermy polynomials are a set of orthogonal polynomials and that's like orthogonal vectors again you should think of them as heroes .period certain ways and if I want to approximate the the distance to go
somewhere to get somewhere I'm much better off having an amount that so long as certain direction that I know for sure like east west and the north south rather than sort of random directions where the depending on which 1 you go 1st the other 1 may be different because there are in fact aligned along each other so the natural polynomials 1 x x
squared execute they're easy to write down and you can try to fit functions by using more and more of these terms but the problem is the EC squared an annex to the 4th sorta look like each other so the speaks so and this idea of things being orthogonal they are 2 orthogonal they actually are sort of along the same direction and that makes it much more tedious than to work out the expansion and to stop at a certain point where you know for sure that you have a certain accuracy so usually we don't use polynomials like that if being clever to even set of functions we use some orthogonal set polynomials instead where we can get a better guarantee the accuracy we can explore some of the properties of some of these functions In a problem that you'll have to do where you actually use different sets of polynomials to fit some unknown function which is not a polynomial and you can see them and the kind of accuracy that you're able to achieve using a certain number of terms just picking them out of a hat or using this orthogonal said and what you what you will find if you do the problem correctly is that by the systematic approach you can guarantee the accuracy and if you just use some set of things to fit but you may not be able to guarantee the accuracy in fact May get into trouble numerically where things become quite unstable sort of like balancing a pencil on its tip you can make a slight error in 1 of the terms and that that wrecks all the other terms and makes it even if you do with the computer somewhat fiddling to work with Of course the thing about these energy Eigen functions years there they don't move in time they just sit there because when we actually put time and time is just a phase factor and the face factor when we introduce it on both science star always cancels because sigh star always has the opposite face factors this one's EDI data this one's you minus I think that's 1 and it goes away therefore hours notional picture of what a harmonic oscillator does is quite different then these energy items states and many macroscopic oscillator that we see is not for sure In an energy and state therefore you solve this problem he had these energy can states and then you might be left asking the question well how this the harmonic oscillator actually oscillate and the answer is that you should think of these energy Oregon State's although they said there and the probability distribution stays the same they change color and color is important to think of because we can use color to indicate the shades and there it we see the ground state harmonic oscillator let's say starts out quite and then it slowly turns blue still sits there but it turns blue and then it turns clear and then it turns red and then it goes back to quiet and so forth and so it is changing along with the real on imaginary part and then minus itself that's fine because square it's the same and then comes back and then does
that and by associating them with the color we can get a picture of how fast time is actually moving and what we would find is that the different energy states of the harmonic oscillator are changing color at different rates the bottom line is playing a low note just going around and as we go up they start going faster and faster and faster and although each 1 is not doing anything if we have an unknown wave function which is a bunch of them added together then we have to add up the colors 1st before we square and their 1 1 or all of it happened be read on the left let's say we're going to get a big number there of half a red and half a blue on the other side we had were going to get a small number there and so what will find as we have this long that's sitting there and it's moving along because of the different rate of change of the different energies and therefore I realize later that we might actually observed is definitely not an energy Oregon State but it's in a superposition of energy I can't states that the energy Oregon State's a very foreign the and situation in terms of our normal experience In the language of quantum mechanics them classical oscillating mass has been prepared in a coherent state it has a mixture of energy Oregon State's their chosen just right so that they form this long and the long goes back and forth and back and forth and so on and if we choose and funny that we get some of the things that doesn't behave like an oscillating mass on the spring at all well of course and quantum mechanics we can make all sorts of funny things that don't seem to behave like anything that were used to because we don't know actually where the article it is until we make a measurement and so it can be appear to be in 2 places at once if we picked the faces to be some funny combination Wikipedia has excellent animations of the harmonic oscillator and I think the best thing to do is to just go to the URL which 0 giving on the next slide and look at the way functions look at them changing color they have red and blue and fact for real imaginary and look at when you take a mixture of them how you get lot that moves back-and-forth and repeats and it's quite an interesting of animation very very well done and I'm so happy that things like that are on the Web and that you can access the it's much much better than the text books that I had as a student where every image was static and it was rather hard to penetrate the mathematics and try to understand what was actually going on and make a connection between something that I understood and this weird set of energy can functions but here's the URL Wikipedia . org wiki quantum harmonic oscillator can go to that link have a look at that and you can get a very good idea exactly what's going on and they have a coherent state and in year really does mimic exactly what you would expect it moves in a nice way very very beautiful and it's very hard in registry with what we understand to be Howell particle moves and in a more advanced courses you may get to the point where you find that you can make a coherent state by taking the particle and just pushing it 1 way and that to push it 1 way turns out will move all the faces of the energy and just such a certain way and I'll give you this beautiful long but goes back and forth that squares exactly with what you do in a classical situation to make a spring oscillate you stretch the mask this way you would never try to stretch the mass like that if you could break it apart because that would the right effect the probability than for the position of the oscillator where the atoms on the ground state is nothing at all like what we expect from classical mass on a spring it just seems to be sitting there and it doesn't say that the turning point it sits in the middle by far much more common to be in the middle and the ground state because of the slump function than to be a the edge but a lot of experiments have verified that that's exactly the way it works and experiments where we use a laser to excite things we oftentimes drawn along there and we say the atoms are most likely to have this separation and then 1 weeping an electron off if we assume that's fast all we did was change the force constant so now that the minimum instead of being wherever was before is moved out if the bond gets weaker so the atoms is still sitting there but now the compressed and the compressed just like a coherent state and so the thing oscillate back and forth and we can get a lot of information on the strength of bonds and Onslow chemistry and very important reactions in the atmosphere by studying molecules like ozone and CO 2 and so forth in the laboratory In controlled situations and working out exactly what wavelength of light they will absorb how long they will stay excited how they will admit the light and so forth and so on and that's basically part of the field of laboratory atmosphere chemistry is to work out things exactly like that basically amounts to doing physical chemistry now we can calculate some of the expectation values like we did for the particle a box for the quantum and I will do 3 of the 4 that you need to do the uncertainty principle and I will let you do the 4th 1 then as a problem to verify that you understand what you're doing we can get the 1st 2 pretty easily the first one for the ground state if I take ax and then I have eaten a minor 6 wary of minus X where I don't need to do much calculation there because he needed the minus X squared square isn't even function it has the same and backs
makes this side negative on the other side positive and I can see right away that the area of that is
sequence of some of terms he is equal to the 0 plus the 1 plus the 2 us so long and it and it could go on forever but we certainly won't take it out that's all he 0 is the solution we have before where introduced the perturbation age super once we have that we have a model Hambletonian H 0 we can solve them and we know all their energies for that case and then our introducing this new new schemes even 1 for reasons that will become apparent in the 2nd is called the first-order correction to usually the most important unless it happens to be 0 In which case and we have to compute to the second-order energy correction however we hardly ever unless we're on track approval .period or have plenty of time on our hands or a specialist in looking at these things we hardly ever calculated beyond the 2 because the formulas get very very big very very long very hard to understand time-consuming to compute and if we can get a result by 2 we may think what we are going to ever get a good result anyway so even if we calculate the use of the term if they start getting bigger we have a big positive 1 and a big negative 1 and the big even bigger positive 1 so for them that's going to be extremely difficult to make sense of where we stop our answers to bigger too small and may be increasingly too big or too small even all the terms if we could get all of them together would would add up to the correct image and likewise we suppose that the wave function can be written as the ideal way function for the model potential which I've written here is size 0 plus a correction the wave function sigh 1 plus another correction site EU and as I said sigh zeros the exact solution which we assume we have and then we have a first-order correction to the wave function and a 2nd or a correction the way we do this you have to do this in a systematic way or you get completely lost on the way to do it is to introduce some dimension less parameter enter to dial into this parameter From 0 1 of the parameters 0 we have the exact solution when the parameters wines we have the solution we want and then In between what we do is we try that connect the 2 solutions like having a steady string between them that we can follow this trail of breadcrumbs From the exact solution we've got to the solution we don't know what it is but it's out here somewhere and that in addition by introducing this parameter it lets us organize all the terms because we can organize them in a power series in this parameter which is conventionally called land for reasons I don't understand but anyway that's the conventional notation therefore what we're gonna do is we're gonna write H is equal to age not for 0 plus Lander Heinz H 1 and now when land is 0 we have our exact solutions when land is 1 we have the solution we want and with the way we approximate things then is to associate with each superscript a power of land if the superscript is 0 it's landed to the 0 anything that the 0 it's 1 if the superscript is 1 it's landed to the 1st power which is landed the superscript is to its squared and so on and then we take our shorten your equation can we say we've got this equation which now has the same land on both sides and if it's going to be true for all values of land the only way that can happen is if separately the powers of land the coefficients for the powers are all equal and then we get rid Orlando we got landed square on both sides are landed at 1st power on both sides whatever we can throw that out then and say Well these coefficients here if this is to squared and this is whatever this whatever has to be too otherwise it wouldn't be true for all values of land and that let suspended a sequence of equations that we can crank through mechanically and get the answer that we want here's how we introduce our parameter lambda it starts to look at a little bit intimidating but it's not too bad we have each side is equal to the and H. now is H not plus landed H 1 side is sigh 0 plus land sign 1 plus Landis squared tied to possible and that's equally side which is the 0 plus land the 1 plus land worried too times size 0 close slander and so forth and what we have to do is collected powers of landed the
1st power of Landers landed to the 0 the 0 power and the that just gives us what we started with if we set that if we look at the terms that don't have any land we have H not sign is equal to sign up but that's what we started with that's what we assumed we could solve as no big surprise good now let's look at the 1st power of land because that's the first one is going to give us a correction on the left-hand side we have h not times so I 1 because that has a land plus H 1 times size 0 where you can see that the sum of the scripts should be 1 and on the right-hand side we have Edie 0 times so I 1 plus 1 times sigh 0 that's an equation now that has to be true and the thing we want is 1 we don't care about H 9 and H 1 what we want is the 1 was the correction to the energy level based on introducing this perturbation H 1 and that's what we want a cigarette Our strategy then in order to figure out this equation because it has operators on the left-hand side and wave functions and it has numbers on the right side is we've got to get it down to be numbers on both sides and it would be easy if in fact the the wave function more nite and function of the perturbation if that were true the perturbation would just be so easy we wouldn't even bother doing this theory and so we can approach the problem that way what we have to do is get down the numbers by a trick and the trip that we do is we 1st take our equation and we multiply on the left by some functions and then we integrated and that way we know we're going to get a number because we know if we take any function and then the Hambletonian away functions were going to get a number other than the expectation value of something or other and that is then going allow us to figure out what this energy is this strategy gives us the following set of equations on the next slide which is coming up on the left we've got H not sigh one-plus plus H 1 so I'm not on the right we've got he 0 sigh 1 plus the 1 sign of what we do then is we multiply every term by sign-on not star and then we slap integral because of those at a equal 1 we multiplied by sigh not star there still equal and while we integrate them they're still equal and that gives us the following equation but the size .period not it's not sigh 1 DX plus the inner gold sign not star H 1 side not the ax is equal to the terms In energy which have signed on site 1 now the last term just has signed on to star the 81 so I'm not but the 1 is just a number and the year-ago sign not star sigh it's Sinai is 1 because the ground the model wave function is normalized and therefore we can pull the 1 out and that last term on the right is just the 1 good because the ones what we're going to try to get and so organist Gooch all the other things to the other side and set them equally the 1 doesn't appear anywhere else the Hambletonian is mediation and that lets us figure this troublesome terms such the angle sigh not star . sigh what we don't know what age not dusted so I want because we don't know what's I want it but what we can do it but we know it's not just a sign not so if we can Fidelis formula around so that sign knots on the right-hand side then we know what we get and we can do that because it's not his her musician and therefore we can set this up as I've done in this equation the where we swapped Chai Wan star Ronaldo H. North Sinai the the whole thing star and that's because the the Hambletonian the energy operator is her mediation and has realigned and I'll use then we know page not site not is he
not because we solved that now we've got a number so because now all we've got is the inner goal of cyanide stocking up 1 and therefore that is exactly we still may not know what that is but we don't need to know what it is because thank goodness that term appears on the other side exactly the same terms so whatever they are if they're on both sides of the equation we can toss them out and then least just the other terms left over so we have won on the right-hand side is the term left and the other term left is the inner goal of sigh not star H 1 if we can do that integral we know signed on we know H 1 we have to have a formula for what the perturbation is what is it doing and we know how to do it girls we can look amount we can calculate them numerically whatever it takes then we can calculate a number 31 In that we To the energy he 0 and we figure out how much the energy has shifted on account Of the perturbation being added it's a really important result but the first-order energy correction is just the expectation value of the perturbation using the unperturbed wave functions that's the main take-home message taking unperturbed wave functions put the perturbation in sandwich it between the the sigh not star perturbation sign on integrated whatever that ended goalless that is the correction to the energy and whenever you all you have to do is an integral believe it or not that's considered a trivial problem these days and many areas of math you have some terrible problem to do and if you just get it down to integral and the man that's all that's left to do then he consider itself you always consider the consultant integral and you don't even pursue it any further he said well that's that's a solved problem course chemistry we actually have to do the angle and get the number but in Matthew may not be may 1 show that something is true or not true once we consider that we know the 1 we can go on a calculating to and get the perturbed wave function and that the second-order energy corrections usually we don't do that on last iii 1 happens to be 0 because if he won happens to be 0 then we don't know what the correction it's it's rather like approximating the sign function if you say what is the sign of a small angle theta if if you if you're a beginner you might say it's 0 and that's true if it is very small that's close but that's not considered good enough what you really want to say is 1 thing at a small sign is equal to the so you have to actually get a correction 2 0 which is your starting point or you don't consider it done anything and likewise if the first-order energy correction is 0 then we have to compute the second-order energy correction because we wanna know what the correction is otherwise well we don't really know what the perturbation has done and will do some of these calculations in the actual probably do on perturbation theory which will give some some practices that but let's vote just close by looking at a model system the particle in a box and apply a perturbation In this case what I'm going to assume is that the box instead of being from 0 to L is from minus over to L over to the same thing but just moved that changes the math but it doesn't change anything else now we have to have the wave function vanishes minus over to and tussle over to as for the ground state you go work out that instead of a sign you end up with the co-signer in fact for the arbitrary matter on the formula now rather than just being assigned function each time is the co-sign and sign as the co-signer minutes aside and so forth even even odds rather like the harmonic oscillator and many other systems the less practice problem here practice problem 10 let's consider perturbation TAX applied To the previous energy levels for the particle a box before we said the potential energy was 0 in the box now organization of potential energy goes up on 1 side and down on the other side like we've got a
slant In the box the question is what's the correction to the energy in first-order and the answer is the correction is the expectation value of the perturbation sandwiched between the non-state the grass days co-signer the perturbations kayaks we put in the In intervals and we just get the integral of coast where X overall concepts between symmetrical limits and because the anagrams and not function the correction is 0 I said that a bunch of times when the is on the correction is 0 and it's easy to see that pictorial because of its odd that it's negative
and the area's negative on 1 side and positive on the other side of vise versa and in any case when the limits a symmetrical the area 0 but I want to show that to you 1 time and so will kick off the next lecture by just proving symbolically rather than with a pictorial argument that the integral of an antisymmetric odd function is indeed 0 and that'll be that'll be an interesting little trick that's hopefully useful for you to OK we'll stop there
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