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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 dummyvariable 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 soforth 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 .

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