Lecture 10. Particles on Rings and Spheres... a Prelude to Atoms

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Lecture 10. Particles on Rings and Spheres... a Prelude to Atoms
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Lecture 10. Quantum Principles: Particles on Rings and Spheres... a Prelude to Atoms
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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: 0:02:23 Particle on a Ring 0:16:49 Quantization 0:24:23 Preparation of Atoms 0:27:16 Spherical Polar Coordinates 0:31:18 Particle on a Sphere 0:33:03 The Legendrian 0:35:06 Spherical Polar Coordinates
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welcome back the chemistry 131 a last time we talked about more realistic oscillators we talked about the 12 potential we talked about the Morse potential and then we got away from one-dimensional system systems which is 1 variable which we're calling you the extra or whatever and we get a particle in a box in 2 dimensions and we
discovered that under certain conditions there was too generous see if the lengths of the box were the same and so forth and 3 dimensions is the same and the main weapon we had was that we took a 2 dimensional equations that had 2 variables and and we separated into 2 one-dimensional equations both of which we already knew the solution to and we just substituted and the solutions 2 days rather than talking about park on a box I wanna talk about a particle confined to a ring 1st of all what should be an interesting thing to solve and particle confined to the surface of the sphere and both these things will be important as a prelude to Adams but both of them are completely artificial because if you say Well I've got a particle confined to a range if -dash yourself what kind of potential does that have that the particle stays just on a circular rank and that's not very realistic-looking potential that if you just get off if you're on the ring that you're just there with kinetic energy and then if you're off the ring by Absalon its infinite or something like that and so we could get some kind of anomalous looking behavior but really what it means is we're gonna set up the equation in 2 dimensions and they were going to freeze 1 dimension and solve the other which will be an angular variable and will leave the radius of the rank as something we fix at the outset and then we go from there and will leave the radius of the sphere is the same thing and luckily what we do real Adams at least for simple ones where we don't have too many electrons was just 1 being the right amount
of we can actually factorize the thing and so we can use the solutions for the ring this sphere but and then just pasted together look exactly like an onion like growing and onion and shells we argued early on that a particle confined to a ring which we can think of is sort of like an electron and classical or orbit I would have to have a wavelength but said and the idea is that every time it goes around it has to match up perfectly because of it doesn't it cancels out and in fact there's nothing left for the probability because it's just basically minus itself half the time and so that was an argument that the 2 broadly wavelength had to match and are qualitative conditions them is that we have to have an integral number of wavelengths and Orlando as to equal the circumference of the rain which is to buy are if that condition is met that we can go around like that and we can come back and will be at the same place and that means that we've got a stable standing wave of probability pattern is not changing In space at time not that the broadly wavelength is related to age divided by the momentum therefore we can just substitute for that and time stage upon P is equal to 2 pi are and then we can write that in a very suggestive way we can divide both sides by 2 turn the agent arranged by and we can all by both sides by and then we we have an age bar is equal to our times that's interesting because ah times PE is going to have something to do with angular momentum and if we have a particle on a rainy it we automatically think of angular momentum whenever we've got anything going in a circle are confined like that we think of the angular momentum of of such a particle we learned about that and classical physics let's have the ring oriented in the XY plane then
that means but the particle as an angular momentum vector J is equal to are that the vector cross product of P in this case are points to the rain and is the way the particles going so on are always at right angles in fact J is equal to are Cross P points in the sea direction if we've got the particle in the ex-wife planes and so we can take this component of the animal manner and related to the particles and that means that we've got which turns out to be the times P time signed data signed dated and signed 90 degrees signed pile to that's 1 the JC Is and that's legal NH bar by the condition that the interval number of wavelengths fit and now sort of by this backdoor route we have that angular momentum seems to be quantified photons Cayman units each new angular momentum comes in units of H bar now I time-dependent showed your equation for the particle honoring Is the following as the kinetic energy With respect to ax plus the kinetic energy with respect to wine 2nd partial derivatives same thing as a proper article two-dimensional boxes EPA times simex wife and the problem is X and Y in this problem are very very very awkward variables and that's because our is the square root of EC squared plus y squared and so on and so forth and if you just tried it just bowled bully your way through this equation using Cartesian coordinates which were set up for square problems but you'll never get anywhere what you 1st have to do as you 1st have to change your variables so that you're in the same kind of house mirror as the problem areas and then the problem will seem very easy and that's what we're going to do we've got a circular problem for gonna use polar coordinates we're going to set up 2 new variables rather than X and Y we're going to set up are like Kazaa constant that's the perfect thing to have an awareness derivative with respect to R 0 0 Dolores constant and then the other variables just where I am on the rain what's called outside that I have a relationship that X is equal to our times co-signed find that this leg of the triangle and why is equal to are assigned finalists that violated the triangle intercourse are swears sequel X squared plus square which is always constant now we have to make unfortunately a transformation not only of the variables x and y but the derivatives and that's trickier so we need some actual multi variable calculus to do their transformation and I'm not going to go through the transformation over and over and over because it gets very tedious to go through it but I want you to see 1 time exactly how you do it so that you'll understand where these terms come from I to do this then here's what we have to do we have to use the change rule from calculus and we have to understand that if we have a function of more than 1 variable and I change something I want figure out the change I should figure out the slope but it does with respect to the ax directions if I change acts and then the slope does with respect to the wider action if I change wine and that'll give me the total change in the function I need to take to things and add them up and for each 1 of them if I use a different variables I can use the chain rule so here we go the derivative of side the wave function with respect to R has to the side the X times TEXT are you can think of these things just like fractions where users cancel out the VAX as I can remember the chain the PCI instincts IDX TEXT that's the 1 but I have to it's a function of 2 variables so I have the PCI BYD widely and now I have a formula for a wide with respect to ah and so at what I get is I get keeps IDX co-signed class deep the Y signed for the 2nd derivative Is the derivative of this thing and it gets messy fast but it's a really good exercise and thinking clearly because what you do is you just take each of these terms and called the side DX some other things and then put that in the formula and then expanded back out very carefully annual seeds don't skip any steps let's take another derivative the 2nd year of sigh with respect R Squared is just d by the art of what we got Dietz IDX cos fly Deeside the fine and I think I can put that in by factoring out a co-signed fight and get the 2nd derivative of outside the EC squared EXT plus a 2nd derivative of sigh with respect X and then with respect to wide the white yacht and you might say have what about the order of the X and Y and the answer is for wave functions and nice things that we deal with any kind of function we have we don't really care about the order of we take the derivative partial derivatives with respect to Why 1st or the partial derivative with respect ex-first Bowser equal so we don't worry too much about that and then we have signed 5 times again to terms both of them were the 2nd derivative with the DAX Yardy widely or if we add all that up we get Coast squared Kansas 2nd intervals sigh with respect X work plus signs with 5 times a 2nd round above sigh with respect Why squared plus 2 signed ficus 5 times the mixed partial derivative the derivative with respect to fly I'm going to leave as a problem we do it the same way we start with the size of the 5 and we either as the chain rule and then whenever we have X over ah or something like that we expressed in terms of co-signed find signed 5 and what you'll find is that you get a little bit longer expression here by about the same length but you get miners are times decidely the AA plus our squared times 3 terms which look very similar to the other 3 accepted with a slightly different ordering and now we can get the relationship we need between the Cartesian and the polar 2nd derivatives because now we have the 2nd derivatives with respect to R and with respect to fight and we know what they are in terms of with respect to on the effects and why it's still quite a bit of algebra and again I'll leave that for for you I'll "quotation mark the results so you can see what it is but there's still quite a bit of algebra to do and what you find out then is the the 2nd with respect to side with respect at square plus the 2nd reading sigh with respect Why squared is equal to the SEC Nunavut of sigh with respect our squared plus 1 over are times decidely plus 1 over our squared times the 2nd year sigh with respect to fight with respect to the angle and we can check that we haven't made any mistakes because we can put in units these all have units of length squared on the bottom forget about the wave function units for the time being we have 2nd derivative D Our square on the bottom 1 over R D R 1 over our squared the firefighters and the length of the angle is just a ratio of things because it's in and now at this point we make the totally artificial assumption which is say Hey particles are ring let's just freeze are and so where we see a derivative with respect to ah let's just say
0 because ours not changing and that's great because now we just have 1 over our squared times the 2nd riveted with respect to find that's looking except 4 fights instead of access looking awfully similar to things we've already done therefore the equation simplifies to this instead of having the Cartesian coordinates we have minor stage square over to an times won over our squared times a 2nd derivative with respect to fly of site which is not only a function of 5 because ours fixed it's a and that's equal at the Times side "quotation mark this is beginning to look really good and here's why we know that end times our it is the moment of inertia of a classical particle orbiting around or a bit of radius r then I were doing a rotational problem by keeping the particle honoring and we found that we just got the rotational constants that moment of inertia coming into the problem just naturally just by the way it worked out the differential equation if we write it in terms of the moment of inertia times the energy is just the 2nd derivative with respect to fight physical the minus ii upon a spot squared times 5 times sometimes and that the solution of immigration like that not surprisingly is an exponential so we can write down but the solution and we can write down look it's a I knew fly was B E to the minus minus-5 new fly use the square to ii upon each part we can verify if we show that in that we solve this Schrödinger equation we get the answer we need to have a high because we have a minus after we take
the derivative twice we get minus that could be I squares minus 1 A-minus squared go round the other side to minus 1 but we can't have any real exponential since so these these functions there are things that court screwing around 1 way or the other and when they corkscrew around they come back and meet and they could go the other way but they meet the quantise Asian arises because the wave function has to meet itself on the way around and that means that if we add to pot the wave function it has to be the same thing if that were true the wave function would wouldn't be single valued in space we couldn't depending what variable we picked college it would have a different value and so but at the same point on the range the the rest of the same value so that means it passed exactly come around and that means that new times to apply physical the too in so the end is an integer so if we had to fight to the angle it has to be 2 pi times a manager and conventionally rather than using anchors and is used for other things to do with energy we used and where M is an integer and end this call the magnetic quantum number wise it magnetic because if we've got a charge on a ring and we imagine is moving around a charge honoring them is a current loop and a current loop makes a magnetic field that's exactly how you make an electromagnet you take a tunnel wiring you wandered around the core and then you could put some current through it and you can pick up all kinds of stuff and that's quite a fun thing to do when you're in elementary school and I remember spending considerable time doing exactly that and seeing what I could and couldn't pick UP In fact we can make another connection with classical mechanics so the angular momentum els e is just ah crossed P z and we can put our quantum operators we can put an X had PY had minus white-hat PX that that's a disease component of our cross P is if we set it up and that's equal to minus ii H bar times the derivative with respect to why minus Y the derivative with respect to act so are operator and we can take that and we can convert that 2 polar coordinates by exactly the same tricks is what we did before and if you take that particular combination and you're very careful when you convert polar coordinates you find out it comes out to be this real simple things minus ii H bar d by the fire just the 1st derivative with respect to the angle that's what we get well that's really interesting because what that means is that but when we did the energy we said what we could have the 2 the I M fly he did the miners I am and we could have a of that must be of that we could have any amount of each but if we want the particle to be an enlightened state not only of energy but of angular momentum then that means it's 1 corkscrew it's the ETA the plus science going 1 way or another the minus and fly going the other way and so what we do when we set up these problems for neatness is Weidar set a equal to 0 and say that it's negative or we set equals 0 we says positive them and so am can vary From -minus some value to plus some value and anywhere in between including apparently 0 and the interpretation is that the energy sequel because particle going this way and some raid particle going the other way but some rate while they have the same Connecticut Energy this then is our final solution he did the I am flying embers of any positive or negative integer and apparently 0 could be
0 why not that solves it as well and that certainly doesn't have a problem if there's no twist at all this just flat course it meets up if we normalize the wave function over over the ring that means that the probability that the particles somewhere on the ring is 1 if we do the Annagrove since the the I 5 times minus and fires 1 year goes to Part we don't integrate over Arkansas last nite in the problem any longer it's fixed just integrated over 5 then we can get our normalized and wave functions for the particle on the rain 1 over the square to at times to giant the lowest energy here is an is equal to 0 which is 0 and this seems to run counter to what I've been saying which is that whenever you have a confined particle you should have some 0 . energy quiet because you want to satisfy the uncertainty principle and the reason why that this seems to violate the uncertainty principle this ,comma glossed over in the book but the reason why it seems to violate the uncertainty principle is 1st of all we just threw are out we really had a 2 dimensional problem but we froze 1 of the variable who says you can freeze 1 of those variables exactly like that that's . 1 and . 2 Is that 5 seems to have an artificial range we say 5 goes from 0 to pie but it would be the same if I went from 0 to infinity because it's keeps wrapping around over and over and over and so if you if you argue Well you don't fight could be anywhere between minus infinity and insanity and it would still be somewhere on the then you could have the momentum the 0 of course you have to have the momentum B 0 if you have the energy 0 but any could still have the position In terms of the actual value of 5 billion determinant kind of a mathematical Dodge but you have to be careful if you set up a problem and then impose a constraint that might not be physical say it has to be exactly on the range and then get in a tizzy that something doesn't seem to be quite right because the arguments may be quite subtle at that point the probability density for an energy and angular momentum is independent of the angular variable 5 the probability density is flat and of course it would have to be because there's no reason why I should expect to find the particle more on 1 side of the ring than the other side of the ring when there's no difference between the sides of the ring so it's gotta be flat and that makes perfect sense but if there's nothing to distinguish the 2 sides how are we going to how we gonna tell and quantise again arises from the fact that the wave function Hester matchup has to be single value now the next step is to expand to a sphere analysts fear you could argue as fears about your and that might be a good way to look at it in fact is pilot brings it's an artificial again to assume that the particle can go anywhere on the sphere but can't move radially at all but nevertheless it's a really good steppingstone to getting to the point where we can write down atomic orbitals and figure out what's going on we've got a sphere we know if we write down the Connecticut Energy and because we've got a sphere were keeping it on the sphere we've got no potential energy except this completely arbitrary potential energy that's keeping the particle right on the sphere we have to convert to spherical polar coordinates In accordance system here His is equal we have 2 variables we have fire which goes around the sea axis changes X to Y and so forth and we have stayed which starts here and goes from the North Pole where the data is 0 to the South Pole where the it is please or 180 degrees but we don't go around again because at every time we go here we make a ring like a tree rings and then we go here we make another ring like a tree ranked only go here to the equator we do that 1 and we go down here and we finished down here and if we won around again we becoming all the rings again twice so far I varies from 0 to for 360 degrees around and the other 1 very data varies from suggests 0 2 1 8 we adopt a right-handed coordinate system and that means that X crosswise the thumb points in the policy direction and that's by far the easiest way to do it because if somebody draws a figure were extra shooting this way and why is that way and you try to read oriented in your mind so access to the right wise and the paper it takes a long time mentally it's quite a gymnastic Regis grab your hand and do it you can figure it out and be careful that because what you tend to do when you're doing problems is you have the if you're right-handed you have the pen in your right hand anyone figure out which way some skull and use your free hand and that's the wrong hands this he do Alexander in physics especially you just get marked wrong because you did news right in accordance system and I've watched that happen to people at various stages of my career Oh here's a right-hander coordinate system failure and fire set out and our position is given by 3 numbers are the distance out faded the distance down from the north pole and fight the distance around from the
x axis what's very confusing is that if you take a course in math for some reason that I cannot fathom the variable said and fire and so it favors the 1 this way and Pfizer 1 this way and I think that's because in math when you two-dimensional problems you always Fader and they just like to keep data for the same variable and use 5 the other ones but in physics it's fly around the sea
access and fade away from the sea axes and you have to keep them straight because if you open the wrong book you'll get things backwards and you get a terrible mess that let's try a practice problems let's consider a volume element this is a physical 2 DX the wide easy and Cartesian coordinates it's a little too little Q what's the volume element for spherical polar coordinates so let's take a radius r angle the from the z axis was the formula for the volume well we don't we don't care but because it's always the same here but we do care would say that is and here's why the size of the onion rings here at the top near the North Pole is tiny 1 here Wednesday at a small estate gets smaller the total size of the ring gets smaller and smaller as we go toward the equator the same ring around 360 degrees and fight is much bigger In terms of the volume that is the little hold if we take a little slice and are and so we have to wait the volume elements by how close we are to the North Pole like imagining you can walk around the North Pole in a little circle on you've walked around all possible longitude starvation try to do it the equator it's a very long walk the same thing so here's the beauty of calculus is that we actually draw this thing as a because of large changing and so the inner part is smaller than the other partners like like a little Conan but when we have just very tiny differences then it's like a Q and all we need to do is take the size of the king and that's the beauty of taking very small things DX is that no matter how Kirby something is if you take it's small enough it's a straight line that's why calculus is so great and so we can figure out the distance here out is far signed faded and so the distance along if I moved by the 5 is this distance fears are signed dated the fire and the distance the other way is just park is as the full distance times the fatal and the distance and the 3rd direction is just D or R neither the identify changes this so we can take those multiplied together and we get the volume it is far squared signed dated Dr Dee Dee said that 85 that's important to know because you're going to have been goes to do with sigh and so forth in them and they're going to have DVD because you have been great overall space and you need to know what DVD is in terms of these variables and now you know what it is the exactly the same way but we did before but I don't wanna take an hour to go through it carefully and I'll just leave it to you if you're interested to work it out once we can take the but Cartesian a 2nd derivatives and 2nd interval with respect acts 2nd the respect why plus a secondary respected and we can cast in the following form secondarily with respect or are close to or over our are times the 1st derivative with respect plus 1 over our square Kansas operator I've called Landis square and Landis squared has nothing in it except the and find it has some signs where data seconded with respect to fight and this 2nd term which is written in a very funny way 1 over signed the by the failure of signed the baiji theater unless you're used to dealing with operators that this is written in a very compact very nice ways you don't have a lot of terms but you have to be quite careful when you actually put it on a wave function because it you only put the wave function on the right of the operator you don't start inserting it in between things you just put it on the right and then you go sequentially right take the new respected their multiply by side that take the derivative again and so forth but if if you don't understand the operator notation then you're very likely to get things wrong because you may stickup and wherever you think there's a blank and that's not correct this thing I'll land squares called Lashonda in this very famous the Legendre polynomials and so forth and as I said the operator takes a bit of getting used to only put the wave function on the right to be careful about that don't put a sign from both those derivatives the operator of the lender squared the Lucia Ondrea has but all the angular energy in the Hambletonian because the other operators had derivatives with respect to ah if we freeze Pa we don't allow any change in our there's no energy that way that
means that this thing Landis Square is what we want to focus on and it's just the energy to do with all the possible angular emotions of things on a sphere so let's fix are throw that out again same way as we did with the particle honoring and now we've got this new equation to solve minor bars squared over 2 and March squared times landed square on the way function which is a function of status and fight is equal some energy as a function of the and 5 and if we can find a way functions that solve that I can value equation and the iron values than we know the energy and we know the possible way functions on a sphere and a intercourse we expect that it's going to be quantized and so on because it's a trap thing again and as we go round and didn't find much more complicated this time Kansas to home so it's hard to see but it's got to be similar as what we had before there is no potential energy here the only requirement on the quantized Asian is just that the wave function fit in the end to the space OK let's do a practice problem here than practice problem 14 let's show that the angular Schrödinger equation is separable and what is at me that means that whatever this way function in favor and 5 years we can write it as a product of something that's only a function of the and something else that's only a function of 5 and if we can do that then we guessed that the something else that's only a function of is what we had before because last time we do two-dimensional particle a box then OK what it was a product and the X was the same 1 is what we had before and so fate is different from fired because fire goes round and to pine paid only stops here so we would expect it's going to be so easy is that because they too could be different different fight was still fight should be the same as what it was before and so that we got that even the I 5 stuff will that saves us a lot of work OK here's what we gotta show it's separable gift we can write the solution as a product if we want prove that what we have to do as we have to rearrange the equation so that we have 2 terms 1 of which only depends on the plus or minus some under the terms but only depends on Friday His equal constant and that we make the same argument if we fixate on change 5 if it's a constant enemies of both of them are constant otherwise that would work and that means that we can do a one-dimensional equation for each so let's try the product trial product solution again we use capital accidents let use capital fate of data to some function we don't know what it is but we don't care at this point capital 5 of and substituted and so we've got the Lashonda on this thing and now we get a number and here epsilon this number by minus epsilon is just too i.e. over-age bars square and eyes the moment of inertia that is cleans it up so that we don't have to write a lot of extra terms now if if we substitute this here's what we find we have 1 over signs squared 2nd admitted with respect to fight of this product plus 1 over signed data set of 1st truly respected data of signed their 1st 2 respected the of the product is equal to minus epsilon times the product I do the same thing I divide both sides the products and 1st I say hi I've got the product of 2 things secondary above of with respect to fly of state some fraction of the data In a function of fire the function of fate as a constant so I can pull it out because it doesn't matter where it is that's what the partial derivative means In the 2nd term where the derivative has risen respective Fader I can pull the function big fire out in front as a constant because it's just a bunch of derivatives with respect to the data and we treat Pfizer constant so let's call it out just like the same way we pull out a constant in a derivative and if we pull those out in front that makes it much easier to see now we've got big fade out in front signed data Square is 2nd with respect to fly squared and then the rest but you can see is able to minus epsilon times the product and now if I divided by the product on both sides the data goes away and the term with and the fight goes away and the term with data but we still have this sign square so we have to multiply through the whole equation by signs were and then if we do that we finally find the following we have won over 5 times a 2nd derivative with respect to fight that's 1 term plus a bunch of gobbled the goop but it doesn't matter what it is because it doesn't have 5 it's all faded and it has an epsilon signs were there that's equal to 0 that's good enough because I have this thing over here which is just and this thing over here which is just there and so I've got to equations 1 of them just the other ones just stay there and I I'm I mean and we can switch once we've got it down 1 variable we can switch to the regular D the start of the funny indeed because it's the same thing we can solve the differential equation so the 1st term as I said is going to give us exactly what we had before b in the in the park along a rank because this same equation basically as the particle honoring equation is that what it's actually going to be will depend on what they happens to be but that doesn't that doesn't bother us we have either the eye and try and then whatever fate happens to be it's going to be some other function that's gonna be are business to solve In the end the 2nd part of the equation which is a a different equation to do and so we can we can break this very complex problem up into this series of problems and solve them and it turns out for a real problem with an Adams organ dues were gonna 1st break up the problem into the article on a sphere and there were just gonna let RB the other variable and not surprisingly were going to try a product worthiness AG I think that the wave function is a product of some function of time some function of from some function of 5 and see if that doesn't work then that gives us a clue that of how to factorize the thing and see that it works 1 will that fail well if it'll fail right away unfortunately if we have 2 electrons because if we have 2 to electrons then ,comma where each of them happens to be and their repelling each other in their attracting to the nucleus it gets too difficult we can't separate the equation and so we run into problems in that case what we do is we treat each electron separately solve it and then take the electron electron repulsion part but we couldn't handle and we couldn't separated and we treat that as a perturbation but how good that will be will depend a lot on how close the electrons are getting you can imagine that if the electrons happily get pretty close in space that I the potential could get quite high and so the electrons will tend to avoid each other and that means that the emotion is correlated the SAR like a cat chasing its tail 1 1 starts
going this way the other may go that way and so forth and so the electrons and may not be independently moving around and that kind of electron correlation is a very important aspect of multi electron Adamson higher dimensional systems but will touch on that and later on in the course and for now I will leave it there please don't take time can look through but these transformations and spend a little time with in a quiet room with a pencil and a piece of paper and just methodically go through and take each step and at each step say What is it me what am I doing why can I do that and go through it and see if you can figure out why these things have the structure they do b d by the fate signed data stuff with the 1 signed the 1 over centered on front that's going to be a little bit tricky to get but you can get that too if you work on and next time what we'll do is we'll pick up on our solution we can guess the solution and fired but we can't figure out the solution and there yet because that's a different differential equation was signed today in it and we haven't figured out anything to do with that equation yet so that's a separate equation for us to solve will have to figure out what kind of techniques we need to solve that equation and then figure out what these functions are and then hopefully we can get some idea of what these functions on a sphere actually look like so we'll leave it there and then pick it up next time to figure out the actual wave functions for particle on a sphere