Lecture 09. The Morse and "6-12" Potential and Quantization in Two Spatial Dimensions


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Lecture 09. The Morse and "6-12" Potential and Quantization in Two Spatial Dimensions
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Lecture 09. Quantum Principles: The Morse and "6-12" Potential and Quantization in Two Spatial Dimensions
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Shaka, Athan J.
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UCI Chem 131A Quantum Principles (Winter 2014) Instructor: A.J. Shaka, Ph.D Description: This course provides an introduction to quantum mechanics and principles of quantum chemistry with applications to nuclear motions and the electronic structure of the hydrogen atom. It also examines the Schrödinger equation and study how it describes the behavior of very light particles, the quantum description of rotating and vibrating molecules is compared to the classical description, and the quantum description of the electronic structure of atoms is studied. Index of Topics: 0:01:02 Odd, or Antisymmetric, Funcitons 0:05:57 The Morse Potential 0:18:05 The 6 - 12 Potential 0:27:28 Quantum Systems in 2D and 3D 0:28:41 Particles in a 2D Box 0:39:49 Quantum Dots 0:42:40 Degeneracy 0:44:40 Particle on a Ring
welcome back to chemistry 131 a today were going to talk about more realistic vibrational potentials the Morse potential 612 potential and we're also going to extend our vision into at least 2 spatial dimensions rather than talking about a particle and one-dimensional blocks will extend it to 2 spatial dimensions and see what complexities that involves it turns out for a particle in a box that won't be serious but depending on the form of the potential having more than 1 spatial dimension can make the equations much more difficult to solve when we last left I made the remark that we didn't have to do a certain integral and the reason why we didn't have to do it is because the underground was off or enter isometric and the interval was symmetric and I'd like to just take a 2nd to show you how that works as a practice problems so suppose we consider an antisymmetric function of X that is that the function of X is equal to minus the function of minus insects then we have to show that the integral of such a function is 0 between symmetric limit just the statement of the problem gives us a clue how we're going to have to proceed we're going to have to make use of the symmetric limits because it certainly isn't true if we integrated from 0 to something that has a finite organ also have to use the fact that will we change the size of the function the argument that the function changes sign let's see how we can do this thus the check here and it's a very good trick to remember is that we don't actually tried to show that the unit is 0 that turns out to be pretty difficult what we instead do it is we show that the integral is equal to minus itself and then the only number that's equal to minus itself is 0 oftentimes very tricky things and math rely on not showing that X too but showing that X is greater than or equal to 2 and that X is less than or equal to 2 women act X has to be too and it's a very good trick to remember if you get stuck on Sunday a mathematical problem is to try to so painted into a corner by inequalities or relationships rather than just trying to get its spot on here's what we can do that the dummy
variable that we use in integration the value of the angle doesn't depend on them that's just a notation consideration therefore if I call the integral which is the integral from Max's said area of the function of of x that's also equal to the integral from you equals minus 8 a f of you do you same thing now I just make a substitution that you is equal to minus so I put in -minus sex equals minus a lower lower limit that she recalls minus minus X equals plus a for the upper limit F of Yousef of minus acts D minus sex and then I bring out the minus sign from the D minor sex and I have minus F of X from minus X equals 8 to minus 6 singles plus and then I use the fact that the function as isometric and I change minds of X 2 F of X and then I just say Look if minus X is equal to a that mean speaks to me of minus accessible to minors and amendments is equal air and so I'm really integrated from Mexico's 8 x equals but I know that's equal to minus suffice swapped the women and that's equal to minus on and so by that chain there I have shown that the year-ago on whatever it is is equal to minus and therefore the integral OK let's move on and talk about a more realistic potential the harmonic oscillator Davis a differential equation that we could solve it was harder than a particle in box for we solve it we could with a little mathematical sophistication write down all the way functions for all the excited states in their new infinite number of them but it's not very realistic for a real chemical bond because Adams when they get too far apart don't interact very well we know that for from gasses and other systems we can only have a chemical bond with the force constant restoring the Adams to an equilibrium position when they're rather close otherwise we would certainly be bonding of sorts of things which does not happen but it's not so the harmonic oscillator is a very bad approximation unless we're near the bottom of the potential if we're near the bottom of the potential is quite good as we get out further it becomes very bad and therefore interested in breaking bonds or simulating breaking of bonds somehow it's extremely bad therefore we need something that's a little bit better and therefore the question is is there an alternative potential which 1 is more realistic and 2 still allows us To get the exact solutions for the time independent Schrödinger questions to get the and energy levels and so
forth and so on and the answer to both questions is yes there is such a potential it it's called the Morse potential found after Philip Morris who 1st suggested it in 1929 but did not take long after wave mechanics was invented for people to start working like crazy on this new field and make all sorts of refinements to the simple problems and really going depth and an advanced the field and this can model of vibrations much better than the harmonic oscillator especially for excited states in a molecule the number of bound states in the Morse potential is finite and that means that there is a limit after which the chemical bond breaks and the Adams flight apart that's already a substantial improvement over the harmonic oscillator and furthermore even the bottom part of the world where we haven't gone very far is represented much better with the more source later than the harmonic oscillator and in fact that the energy levels always get closer together with the more source later as we go up this will see OK what's the functional form well it's something that's not very obvious at all it's that the potential as a function of art should be some constant deed with units of energy times the quantity 1 minus need to the minus 8 times AA-minus RE quantities square that's not a very obvious potential to choose and it's not at all obvious what at 1st blush that 1 could solve that potential in the Schrödinger equations and and get solutions that looks to be fairly complex here D is called the well that and Bay is called The with parameters and we have 2 things to control we can control how how deep the well it and we can control how wide it is the depth has to do with the total energy before we dissociate and the way it has to do with the spring force constants on the debt they're related when the game is small the parameter in the exponential what that means is that the well is narrow and there's a larger force constants here's a comparison that I've adapted from Wikipedia on that the harmonic oscillator which is in green and the Morse oscillator which is in blue and there are a couple of things to notice the more source later is this beautiful function which is asymmetric pretty much a very realistic way 4 a real chemical bond the levels get closer and closer together as we go along and there's a definite prediction about what the dissociation energies the dissociation energy it is not well that but it's a little bit different than the well depth because of the 0 . energy of the oscillator we call the dissociation energies the not and the welder D and there's an equilibrium distance which we assumed to be identical for the 2 oscillators in this case are so big the equilibrium distance now back the energy levels for the Morse oscillator but which we won't solve the equations and go through all that and this course but just to quote the result the energy levels of the Morse oscillator get linearly closer as we go over 1 the spacing between 0 and 1 is 1 unit than the spacing between 1 and 2 might be . 9 and then . 8 on points of that equation fits this functional form the energy of quantum level B which is an integer number is a borrower make times to be plus one-half that's exactly the same thing as what we got for the harmonic oscillator except that kept going and going and then there is a new term which it shows you how how clever this potential is because it gives you what you had before and then it gives you just the right correction to make a more realistic miners staged borrow major X times quantity the plus one-half square and therefore there's this quadratic terms that's lowering the latter of states this week clearly there is some value of the where the quadratic term get so big that the thing would turn around and head back down and at that value of the where it starts doing that that's the highest 1 after that the bond is broken and we
don't consider that that energy equation any and the dimensional as parameter excess the is just call the Anaheim inner-city constant it's a measure of the distortion away from a harmonic oscillator and to a different kind of motion that hangs out here for a long time and it's definitely not a harmonic oscillator in terms of going back and forth evenly and as I mentioned at some point the max the energy Of the Max plus 1 according to the formula anyway would be less then the energy of the Emacs estate stopped going on at that point that's the highest level but can be supported in the potential and at the that allows us to count how many states could be in there and get an idea of the wave functions furthermore Sasser later quite a bit more tedious to calculate than the way functions for the harmonic oscillator and the harmonic oscillator was already so a much harder than the particle in a box and therefore were not gonna write down the explicit form of the way functions what we usually want in chemistry is the energies to get an idea of what kind kinds of processes can happen if we put in light of a certain energy will be absorbed could I get this thing to break a bond and so forth and so on occasionally we're really interested in detail and we'd like to have some idea of the electron density but usually it's very very hard to calculate the wave functions of the other tricks like density functional theory to calculate the electron density without actually
trying to calculate the full-blown way functions for real bonds them just building on this Moss theory 1 could suppose that you could have another and ominous a city and in this case it's widely for all real bonds the energy levels get closer together when they're near the bottom and that's why the first-term has a minus sign because chemists like positive numbers so we like exceeded be a positive numbers so we put a minus sign there so that X C is a positive number but why the next term in the plus 1 has to In this power series than in the energy as a function of the Is is could be plus or minus and could it could depend a little bit on how many levels you're trying to incorporate into this equation when you do fitting and that when we have a plus and we allowed to be why he'd be there bluster minus as the case may be In can 131 b what you find out in higher spectroscopy when you do that as an actual method of analysis is that you will find out that I R or infrared spectroscopy is very important method to clean all kinds of information about the strength of bonds and rotational and vibrational states of simple molecules and in fact it's of course the vibrations of molecules like C O 2 that are causing all the problems with climate change and radiative forcing as it's called as the earth emits infrared radiation which you can think of it as he comes out of certain wavelengths some of it may actually instead of going out and outer
space and cooling off like taking cookies out of phenomena putting a pie on a window sill some of it may actually hit C O 2 and excited vibrations in C O 2 and C O 2 may vibrate for a while and then it may decide to emit a photo and when it admits it it has lost memory of all direction and may have spun around and they have done all kinds of things in between is so admits in all directions In particular admits it back toward the ground and what that means is that the rate of escape of radiation is altered and when you change the rate of a state of heat you heat that's all you do when you put on a pocket he changed the rate of escape of heat from your body and your much warmer than if you don't have it off and so if you we study the vibrational modes of these molecules in the lab we can predict which ones for example might be very bad to releasing large amounts into the atmosphere and unfortunately 1 of those is exactly the molecule that they put into car air-conditioners hydrofluorocarbons which have very great greenhouse gas radiative forcing and could be very bad to release the force constants that we obtain when we look at these and the bonds really agree with exactly what we would think would happen when we have a single bond or a double bond a week on strong bond if the bond is stronger than the force constants is is pretty big and if the bond is weaker it's small and here are some examples the force constant heroes in Newton's per meter for hydrogen fluoride its 970 a reasonably strong for a hydrogen iodide it's only 320 both of those single bonds but the overlap between the Orbital flowering and hydrogen is much better than I which is a great big thing with diffuse elephant here orbitals makes a very weak bond and carbon monoxide as a force constant around 1860 when we draw the Lewis structure for carbon monoxide we 3 bonds between the carbon the oxygen so that makes sense that that would be an extremely strong ones and other ones with double bonds in between and so this at least makes very good qualitative sense when we compare what our notional idea of the strength of a bond is with this force constants of this spring in this quantum-mechanical model of solved now there is potential that is used which you should know about and which is of historical importance mainly because of its simplicity and the way computers work and this potential is called the Lennard-Jones or 612 potential is commonly used for example to simulate liquids in the early days of computer simulation we couldn't do complex systems like Oprah membrane proteins and all these other things that we can do now in fact what we started out doing was the simplest things like noble gas liquids try to simulate a liquid argon something not doing too much and see if you can figure out 1 based on the potential when it will solidify when it will boil 1 will do this and that and that can already be an extremely challenging problem with the 18 electrons you have but you can make a very approximate model of this because 1 2 molecules a 1 2 atoms a far apart there is a very decent model for how they will attract if there is any kind of charge fluctuations of the charge card of 1 and it's very unlikely that if you have 18 electrons for example that they're all going to be totally symmetrically distributed every which way at every instance suppose just for a tiny a bit of time the electrons and have 10 on 1 side and made on the other wilderness side looks positive and if anything's nearby that then these electrons may rush over because they may see that there's a positive charge here and so they tend to attract and that means they always tend to attract and when you work out the way it works with them to die polls won over our cue ugandan
attraction the ghostlike like 1 over to the 6 so for a long way away it's flat and then as you come in and starts going down down down 1 1 over on the 6 but we know 1 when they get too close that they repelled and they repel because when the electron clouds overlap Polly tells us that we can't have electrons in the same war rebuttal With the same ,comma number more than more than to it and then even then to have opposite spent so they repel they bounce off and there has to be a strong repulsive force there is not such a good theory however about what that repulsive force should and that but we know it's very strong the potential while it gives this nice theory has 1 thing that is a little bit troubling and that is if you go back and I'll let you do that go back to the formula and you put in or equals to 0 so that the 2 particles that too nuclei we think of them as positively charged are right on top of each other were extremely close I we'd expect tremendous repulsion under those circumstances but the Morse potential gives us some finite values it doesn't go up to infinity but Infinity may be too big but a finite value like the Morse potential may be much too small especially when artists timing and usually we are modeling such small values of our anyway but if we have things hitting hard we want to make sure that they hit hard and that they tell appropriately for trying to model that kind of behavior and the 612 potential instead of having a finite value like the Morse potential we just had a term plus a 1 over art of the 12 and that goes much much faster than minus 1 over at the 6 goes down so we get something that goes down to a minimum and then goes up like crazy when you get to short and then we can adjust by and some terms we can adjust how fast the beat goes down and banned what the man with the equilibrium is for the best situation the minimum energy but so the numbers and 612 refer to exponents the 1 over at 6 1 over on the 12th and that's it said very nice functional form here it is the AMA's just could be a number like a over art of the 12 miners be over art 6 that could be a simple way to put it in and a and B are ya to parameters that allow you to control how deep it goes down and what the equilibrium is for the best the distance between another words how big do I've basketball stew of tennis balls you know what what what's the right scale usually we tidy it up so mathematically like theirs we Sabia virus Epsilon some number with units of energy times are on Over art 12 minus 2 or more hours 6 and and is the minimum and epsilon is the depth the energy minimum that are equals or and as I said RAM is the minimum value where were at the equilibrium where it's most likely harmonic costly for example now white Pichot at 12 1 not pick up something else and the legacy of that is kind of interesting in the old days the Digital computers were so slow that it was very time-consuming to compute this potential and if I want to that simulated a lot of particles moving around and I gotta figure out who's attracting and how much and was repelling that I have to basically compute these the of our over and over and over and over and computing something like an exponential function for a co-signer sign in the very early days that was just a lot of machine code instructions and took a very long time to do and so it's slowed everything down the beauty of this 612 potential is the fluctuations fell like art of minus 6 and then once I got a lot of the minus 6 so I know what that number is I just square it I just multiply it by itself and I get the repulsive part are
to the plus 12 and that's just 1 multiplication and that's why I became so popular because it let you simulate things for a lot longer than anybody else could do with some of the more realistic potential that might involve an exponential function or something else and then it let you do some interesting numerical experiments on the computer it's just amazing how much more powerful computers are today than they were when people were I was doing landing on the moon and and things like that but the computer I used as a graduate student had 8 K memory and I remember quite clearly when they upgraded it to 32 K I thought What am I going to do With all the massive amount of RAM how could I possibly use at all and when we wrote programs we used 3 variables x y and z because he didn't want to run out of memory and because he didn't want around out of this basically never put any comments on because that would take more space on the desk and at the end of the day you had to be extremely careful to note yourself what you were doing because nothing had any name that made any sense it was all x y and z i j k and so on and nothing in the Commons therefore if the program was South and you made an error it could be extremely difficult to debug OK as I mentioned are the art of the plus 12 apparently are the minus 12 Excuse me plus some number of erotic minus 12 has no theoretical justifications but it was just too easy to compute there are other forms and nowadays Lennard-Jones is still used quite commonly because it's still fast but there are other forms and computer power so much higher that it really depends on what you're doing and how accurately 1 compute these potentials what kind of form you use what you're trying to match in terms of physical property OK let's move now referred to a simple quantum systems and 2 D and 3 D and talk a little bit about the generously which is quite important in real systems ultimately to do anything with real atoms or molecules we have to be able to solve at least 3 spatial dimensions x y and z but as a stepping stone let's try solving into 1st and get the math down for that and then we can extend to 3 How easier hard it is to solve these equations really comes down the potential if the potential has a simple form then it's easy if it has terms like X times Wyanet and it can be very very difficult to do usually the best days is no potential that's particle and two-dimensional box sets were going to do 1st nothing can go wrong there and after that some kind of potential that's additive separately in x y and z but didn't have any cross terms doesn't have anything X times a year or anything like that OK suppose we have a particle and is confined to a 2 D region Oh here's what we're gonna say In the potential energy is 0 if X is between 0 and Al and and lies between 0 and 0 1 and otherwise if you're outside the box the potential is infinite we know by our argument of a one-dimensional box that means the wave function has to vanish at the edges both with X and Y otherwise the total energies incident particle cannot have infinite energy now we have a two-dimensional equation luckily and so I XY we don't know what the former site of X Y years but we have just the kinetic energy -minus H. bars square over 2 and a 2nd derivative with respect acts we've got partial derivatives because we have to keep clear what we're keeping constant and what we're letting the variable plus the same kind of terms 2nd derivative with respect to why simex wise simex 1 and as I mentioned we have to use partial derivatives because we have to be absolutely clear what we're keeping constant and not only have both X somewhat in the way functions the 1st thing you always try whenever you have an equation like this even if you don't know anything about it Is you try taking a product the try saying I think the side is on of of X Y is is functional Max therefore the times g of y but that doesn't include that doesn't encapsulate all possibilities by any means but what it does do is it means that the something like this the energy and acts in the energy and why are separate from each other and they're just going to have up separately and so the that's the simplest assumption is that they have nothing to do with each other and fortunately for the simplest kinds of Adams like hydrogen atoms that's really good and we can figure out all the beautiful wave functions and make these beautiful electronic orbitals which will get into later on this course if this assumption that size of products f of x Giants GUY works then it also a problem immediately and you're done and it doesn't take too long to see if it's going to work if it doesn't work that means that you've got a bit of a nasty equation than you consult an expert that's why we have experts there in their offices there with all the papers piled up and they can solve a lot of mathematical problems are you go to a reference book on differential equations are you take a more advanced courses in differential equations and all those things are good things to be able to do so that when you get a problem 1 that you can solve it if you can't solve difficult problems quickly that the thing is you lose your you lose your train of thought and you may have a problem many may say Gee if I it's all this I could figure out this other thing and so forth but if you can if you keep hitting it and you can't solve it you don't know where it's going and you can make progress is like reading a novel letter by letter and it takes you a long time to recognize you've got a word the problem is if you go so slowly like that you can't even get to the train of the story because by the time you get halfway through the chapter you've forgotten what the the plot was and is a little bit like that in these these branches of science if he can't see things quickly enough by learning how to do it takes too long and then you get lost and then the problem is that you don't even clean the chemical the physical chemistry knowledge
out of it because you get lost in this labyrinthine maze of of mathematics OK now if you can't solve it and there's no expert what you can do is you can try to throw away certain things from the shrouding equations that are crossing you up and then you can treat them as a perturbation because at least we know how to do that so we take any nasty turn that prevents us from factorizing and put that aside and then we can calculate a correction based on perturbation theory and that can also be a good way to go if you want figure out something quickly whether something is going to have a certain effect or not How do we know that it's worked well when the 2 the equation separates into 2 1 the equations 1 the equation is some equation that has derivatives with respect acts and blah blah blah XXX sex nothing else and then another equation that has everything with respect to what I nothing else but there's no sigh of X Y left if you can separated like that its work if you cannot then you can have the the treated as a perturbations Oregon expert help but commonplace To use capital X about rather than F of X and capital Y of wine rather than GUY but it's the same thing capital while Y is a function that depends only on the argument why it has not no axes in and is likewise they could have constants of course but they can have effects but now equation looks like this the 2nd that energy operator kinetic energy operator and now rather than operating on simex why it operates on exit access times y of wine and it gives back he times ActiveX comes alive Weiss just putting this in as a and in the partial derivative with respect to acts we can take y of Y which doesn't have any axis as a constant and we can move it through the partial derivative operator and put it out front as a constant and vice-versa on the other derivative we can move the part that we don't have to worry about front and then after we've done these derivatives we now have wildlife times a 2nd derivative effects of X plus excellent extend the 2nd rule y y all times minus a password over 2 and that's equal to the energy now there's 1 more trick and 1 more trick is we divide both sides by acts of acts times 1 of 1 and if we do that then the why of wine the 1st term goes away and we end up with 1 over big acts of 6 and the 2nd derivative of the text with respect X and the other term has won over why times a 2nd derivative of y with respect to what and that's now equal to the some number now the 1st term has nothing to do with what so if we look at the 1st turn it has no lies in the 2nd term as a bunch of Wieseman and so if I changed little y In the 2nd term it's going to be different values and that means that it can add up to the energies of particular value of X and let y move around and that can't be a constant unless X itself that 1st term is a constant separately some constant and the 2nd term which has only 1 eye and it is also some constant and that's the argument that allows us to seperate the equation into 2 they have 2 seperate because when I hold 1 fixed and let the other 1 fluctuate around the answer doesn't change therefore they each separately after come to some constant otherwise there could not be true so the sum is a constant and that lets us write these 2 equations now I've said OK the energy for the part is ease of access the energy for the white part His use of wine and now because I don't have any other variables except accident 1 and wine the other day I can also switch from the funny-looking D for partial derivative to the regular D and I end up with the same equations that I already had which was the equation for a one-dimensional particle in a box 1 among uses the variable backs and is from 0 to Elsa backs the other uses the variable why big deal is from 0 to else of and we know what the solutions are because we work those out big acts of X is the square root of 2 over Alex times the sign of Annex Pyrex over and as an integer 1 2 3 Central and Big Y was the same thing just instead events we use what and our solution then is the product of the Oso rather than 1 sign function this way in 1 dimension we've got 1 this way and we've got 1 this as well and that means that at the corners where they're both small the particle really avoids the corners the opposite of amounts really stays away from the corners stays in the center of the box in both dimensions in the ground state so final solution for the whole thing when we put the energy and so forth I've written out here and gory detail the energy E depends on to quantum numbers it depends on Annex which is the amount of ex citation in the direction you can think of that as some to do with other particles moving that way and then and some which is the amount of energy in the y direction you can think about is how the particles moving in the wild direction which is independent of the acts and that its instead of just 1 it's just age square over 8 m times quantity Annex squared over Feliks squared plus and ones were real why square and the wave function is the product and we've normalized it so that the integral over the whole box in both dimensions it is equal to 1 we can clearly see the effect of confinement by looking at real quantum systems it wouldn't surprise you but if I take a 3 D Bob and I make the assumption that the wave function is big acts of acts times big y of white hands being easier to see and I go through the same the mansion nations I find it works and I find the energy has another term that has and square over LCD square and therefore if I make particles of certain sizes that are otherwise the same but could have an electron in the particle that's basically rattling around a so-called quantum Don but then if I look at it the light emitted From this electron moving between levels like some level and equals 2 2 1 and emitting light which can happen I can get an idea about the size of the box by looking at the light emitted a highlighted the electron to go up to some high-level
work I put in a photon with enough energy to remember with the photoelectric effect we could kick electrons up by using the life of a certain energy we show potassium there if we put in light with a certain number of electron volts of energy per photon we take electron up and so we can put an invisible light UV light which we can't see we can excite electron up that rattles around it comes down to it emits a photon and the photon has asserted frequency and wavelength and if it's in the visible region we can just see it and this has practical applications of here I've shown this is a beautiful illustrations which I've adapted again from Wikipedia that shows when you have so little boxes of different sizes that you get different wavelengths of light and you can actually choose but you want by choosing the size of the box well that's going to be extremely useful if you've got an application for a display and you want the color to be readily you want this event so this this has immediate applications in all kinds of fields this ability to just change the size of the box not change the material necessarily but just change the size by how long you let it go or how how heeded the various other tricks of "quotation mark colloidal synthesis and then just change the appearance of the final thing OK let's do a practice problem on the generously because DeGeneres he comes in and this two-dimensional box let's go ahead and consider it to be quantum box and let's ask the following questions under what circumstances would there be more than 1 distinct wave function with the same energy that's what the generously me it means that there is more than 1 way functions that has the same energy as another 1 but is a different functions usually what happens it is if you've got the generously means that you've got some kind of symmetry if you've got a cemetery in the problems that often times you end up with the generosity and likewise the generously can be a clue that you have symmetry there is also something called accidental the generously which just happens that 2 things just happen to be close but that's not likely that's not likely to happen usually it's symmetry related the let's have a look what what would be a cemetery well if we look at the quantized energy we see it as an annex where over Alex where close and Grover white square we can easily see for example the easiest days is suppose we choose LX and l wide to be the same so instead of a rectangle so we have a square and then its annexes 1 then why to that's the same energy use and access to and and what I wanted and their related by symmetry now this is maybe not the only condition let you puzzle about how you might discover all the conditions and it's kind of an interesting problem molesters have a look at the cemetery what I've done here is plot what it looks like for Allied sequels L Y and to make it more interesting I picked an X equals 2 and then why Wallstreet and I will compare that with N Y equals to an X equals 3 if we plant contour plots where the light color is the wave function pointing up and dark colors the way function .period down you can see that when why is 3 we've got 3 follows up down in the why direction and when an axis is equal to about 2 lobes which is just down on the other way and they're multiplying each other so you get this kind of egg Bangkok patterns of light and
dark areas indicating what were the wave function is positive and negative and if we look at the other conditions where we swapped the quantum numbers all we do is we change whether there's 3 going this way there 3 going up it turns out that that's the same exactly is just taking the whole box just rotating which is a cemetery operations and so we can see that the fact that they have the same exact energy is the fact that you can take 1 function and you can just grab the whole thing and rotated in space and just change what you're calling X and Y the same shape that I had the same energy of course this is a very important point is that symmetrical systems always have the generously and whenever you've got energy levels that are close together that means that perturbation can have a big effect because now anything that changes the length of one-dimensional does something can make 1 about but slightly lower than the other ones and often times that happens so In other cases there may be individual tiles rather than the whole thing rotating for example I could have a situation where subsections of rotate like gears and make a new pattern and that would still didn't have the same energy or could it as so it needn't be such a simple thing just overall thing but it could be some more complex relationship between the things and you can explore the situation for example in which 1 dimension is twice the other 1 and you could have different situation and in 3 D of course because we have this Exeter N. C square over Elsie square now the possibility for Tennessee is higher because now we have a 3rd dimension and if any of them are equal or the multiples on various other conditions we could have energy levels that are close to the same energy so there more chances for deal Genesis so in closing if there is any perturbations to the quantum system it will usually list the DeGette at 30 generosity and what happens is that if we have to energy levels that have the same energy but we only have enough particles to fill them up halfway what may happen is that something may change so that 1 energy level was lower then the other and then both the particles 2 electrons going to the lower state that spontaneous symmetry breaking can cause some complexes to get distorted so that they change from their ideal shape and that's a very interesting area of study for example in in organic chemistry of some complexes and the reason why that happens is just because there's a difference in the number of particles versus the number of energy levels that are available and so of course it may try to adjust moved slightly change the length of the box so to speak so 1 drops down and then that once occupied and once that happens as the lower energy state and it just stays like that I'll close there and next time what we're going to do it is due more realistic two-dimensional problems were going to start with a pseudo one-dimensional problem which would be kind of interest in the particle we mentioned that when we're talking about the broadly wavelength but I want to go back to it and then a two-dimensional problem which is a subset of a three-dimensional problem which is the particle on a sphere and these will be our stepping stones to get up to understanding how atomic orbitals Our formed in Adams like hydrogen so we'll will do that next time around


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