# Lecture 12. Spin, the Vector Model and Hydrogen Atoms

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better dictating where the light comes out if future take differences between them you could work all all the states out it would make sense then and if you get all these combinations 1 over and squared minus 1 over and square that the energy levels themselves go like 1
over and square and could be 1 that would be the bottle 2 3 and so forth and maybe up to infinity and this era of reference here for this kind of atomic systems and all the systems in general is that you take the electron the protons and you move them apart so they're infinitely far apart and you leave them at rest if the infinitely far apart the electrostatic energy 0 and if the rest of the kinetic energy of 0 that imaginary reference state you call 0 and then you compare what the energy use as the yen and of course the energy is going to be negative in that case because the way to think about it is if I'm stuck down here where they have attracted each other and I want to try them back out To here I've got to put in positive energy so as I go the other way from the reference I had to go down in energy I could go up and
energy radiating energy back when I was asked to infinity therefore we can right now the the Hambletonian and the wave function for the hydrogen atom we've we've got the trolling the equation time and then ensuring the equation we've got a clue from the experiment as to what happened and so will write down the Hambletonian and it has the kinetic energy of the nucleus In this case is a single proton is kinetic energy Of the electrons and it's gone the potential energy between sovereign this year chat is minus a past where over 2 times the mass of the electron times and this is the 2nd derivative with respect to where the electron gears minus 8 per square over 2 times the massive nuclear stance a 2nd derivative with respect with nucleus minus the square over for clients along on are where are is a special thing is the distance between them now suppose we solve this just written like this what was the wave function depend on well it would depend at the minimum on 6 quarters because it would depend on where the electron is in excess y and z and then I would depend on where the nuclear centers in x y and z that's 6 things but I have to put
in To get value for the wave function now hardware it's like that it there's no way I can plot that because I can't see what I'm doing it's as if I make a value I have 6 actually all I can do is make like contour plots and make 6 certain values and then cut through just like we do with mountain range would make a contour plot return a three-dimensional thing into a map that's flat we drop counters on but here I have to fix a bunch of the and then led to a variant plot the wave function and it's really
really blinded it's like being stuck between 2 tall buildings and you just can't see where honor if you are in the city because you're just going down this narrow alleys and so this is completely unworkable and as you get more particles involved but the real way function itself is just impossible to understand what it actually looks like and it's extremely hard to calculate what it is as well since the wave function gives us the most information possible most of the time we don't need to know Everything about we just need to know enough to be able to calculate what we want to calculate and there's a trip here I'd like to go through to show you how we get rid of all 3 of these quarters so that we just have 3 things and then we can make these plots and in 3 D space where we use transparency and chased indicate a certain percentage chance that the electron is in a certain area so the 1st thing we do it is are potential depends on just the difference in position between the 2 things so that's a clue that trying to specify the exact position of the nucleus and xyz and the electron xyz is kind of a waste of time and I need a trick so that's what I'm looking at is the nucleus and then I can pretend the nucleus is and 0 and then it's out of the picture and the way to do that is to change our coordinates so that 1 system is the center of mass 1 system is wherever the Proton is in the electron is data the center of mass which is very close to the Proton and I call that began and the other part is called the reduced mass which is should be familiar from vibrational problems and that's given the symbol usually and that's the mass of the nucleus times the mass of the electrons over the mass of the nucleus plus the merits of the electron and when the nucleus is heady new is pretty much the mass of the electron the masses the center of mass just drifts around as a free particle because there is no potential the refers to the coordinates of the center of mass the potential only refers to the difference between the 2 particles not aware they are in the universe as the center of mass and we know the solution for that that's just our plane waves ETI PX upon a bar or and here side for this begin at the center of mass is equal to some constant and speak to the minus side carried of and so that's done now all we have to do is outside a derivative with respect to that that's all kinetic energy all the center mass and have his kinetic energy can ever have any potential energy In this kind of problem the other part which separates but only depends on the difference between the 2 particles not not the absolute coordinates and we can write that Dennis minus a spot square over 2 times New still squared were now Dell is referring to the difference just the difference what's the difference between acts between the nucleus of the electron y and so forth and then this is it .period usually we are interested in the center of mass motion we know that things can drift around in fact we try to design experiments with things really very cold very quiet so that things are drifting around too much because if things drift around we get a slight frequency shifts Just Like a Train Whistle coming toward you were going away and usually if we're trying to do an accurate measurement we don't want that stuff we'd like to see know what the train whistle is when the train station mathematically Though this lets us just forget about the center of mass we just take the coordinator of the nucleus wages artificially totally artificially assume it's fixed we don't can be fixed because of the uncertainty principle but we just assume that it is for the purpose of the calculation and then if were really doing something detail we understand that calculation later now then this is great because we're down to 1 derivative and so on it's a three-dimensional problem but the the potential has spherical symmetry and therefore we just imagine the nucleus and 0 0 0 the electron then is the spatial variable and we just go forward and as I remarked of course and real factor nucleus can literally be fixed we go ahead then we know how to transform directions but the by the expertise the writing Weisberg and so forth into spherical polar coordinates we get a partial derivative the 2nd partial derivative with respect are close to or over are times a derivative of the respect to 1 over our squared times the same thing we had before the particle on a sphere recall that before we get to this point and then we just said well artists 6 let's look at the data and 5 now we've got are in there but we've got the other thing where we already know what that popped up and so all you have to really figure out is what this is part argue where instead of being fixed it has this electrostatic
potential between the 2 particles that only depends on arm but not on the data and 5 and so as I remarked we solve that part of that problem on a sphere no big deal and all we do then is we just assumed like we always did that the wave function whatever it is is a problem and it's a apart in Oregon and call God of and apparent failure and firing just gonna call why because that's the conventional terms for the spherical harmonics wireless and if you can verify on your own is an exercise that if you make this the substitution into the shown your equation that it will seperate you'll get 2 parts 1 part the depends only on day 5 the pots pans only on are therefore you can do them separately and you get the following 2 equations you get my State Park Square over to New Y times what we call Landis squared which was all the signs signed data and deep 5 times twice constant and the other part we get is my State where over to new big are Times Arts where did the squared they got do yards square close to the big art plus V are squared minus the odds were and that should be a minus whatever the other 1 constant ones so that the 2 of them together add up to 0 but their separately constants 1 she picked 1 as a number of the other is the same number with the opposite side we already know the top part is just the particle on a sphere so good good thing we did that we know that spots my sage where over to New Times square why is equal to square over to New Times L. times also swung times y and so the constant h password over to New Times L. consultants 1 that's the constant with and the 2nd equation we can simplify instead of dealing with big ah let's make a substitution little you of our is equal to or times they are and let's see if we can list do this together as a practice problem now and see if we can come to some conclusion so here's practice problem 15 show that the differential equation simplifies substantially if we expressed in terms of you rather than in terms of big are what is the connection with the classical case OK well where can I write you is equal to little art and big on we're going to have to use the product rule to take derivatives so that being the case the unity artists are debate plus are times a derivative of our with respect on which 1 and we can take the 2nd derivatives we take a derivative of the derivatives and we just follow the same thing the 1st thing is part of the squared big RDR squared plus Dr are over little Dr plus Dr over little the are and then we get to terms of the are squared RDR squared plus 2 times a derivative of big are with respect to all that we just saw all 4 the 2nd derivative of big are and what we get is 1 of a R D squared you Dr square minus 2 Dr debate on Dr and if we substitute that into our original differential equations substitute the 2nd round then it turns out that minus 2 debate on the little Lotte cancel the other the other terms that I had perfectly and we find the following ban that we've got this some of these terms plus the R Squared minus-2 yards square is equal to the constant which is my State spots where over to new Times all times all at once after we cancel the terms here's what we get we get minus a spot squared over 2 New times little are over you times little times a 2nd review with respect to our sport plus the odd square minus the yards for a sequel to this cost if we divided by our squared multiplied by you we can simplify that and we can finally write the equation In this form that I've written at the bottom where we have the 2nd derivative of you with respect .period swear that looks like a potential and that looks like a kinetic energy Excuse me in 1 dimension there we have minus the squared times you of ah that's a potential he scored over for clients or and you've that's a potential now we have another term times you of art which is a part square over to New York times Altintop books once this whole thing should equal the times you of ah and that's just a one-dimensional problem now on the interval sequel 0 2 Infinity's
rather than minus infinity to infinity like acts slightly different Zero to Infinity and we can write simply as pseudo kinetic energy operator operating on its way function you plus the effective on you have our and the effective is the real electrostatic potential which has won over our dependence plus H spots where L and selfless won over to new or square and that looks just
like a centrifuge centrifugal the term 5 something moving in a circle it has an energy elsewhere over to I and that's exactly what we would expect if if we have non-zero Bell Bush has angular momentum that we had to have the energy of the angular momentum before we solve it for you but that's the nice simplification we still have to solve for you and then we have to sulfur are in order to see what kind of electron density we would expect to find 4 4 as a function of little are which is the separation so that's what we'd like to know is we got a hydrogen and where the electron hanging out and what's the chance of finding it here here and so forth In the case of 1 palace not equal to 0 it is harder for us to do why because what we had that kinetic energy terms with the potential energy 1 over are well it's tough to figure out about that because last time we had no potential energy for the particle box or one-half kx squared for the harmonic oscillator now it got something a little bit more challenging 1 are in particularly might I worry a little bit but when are 0 words that blows up and so maybe it's going to be very difficult to solve differential equations it turns out that's not such a big worry and now we've got another term this at all times all close 1 upon our square and that's about as another thing is so malleable 1 over on 1 over a square and we have to find the functions that does that where can I explore that how do that step by step in the next lecture were all talk about these radial distribution function
Lecture/Conference
Lecture/Conference

 Title Lecture 12. Spin, the Vector Model and Hydrogen Atoms Alternative Title Lecture 12. Quantum Principles: Spin, the Vector Model and Hydrogen Atoms Title of Series Chemistry 131A: Quantum Principles Part Number 12 Number of Parts 28 Author Shaka, Athan J. License CC Attribution - ShareAlike 4.0 International:You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor and the work or content is shared also in adapted form only under the conditions of this license. DOI 10.5446/18890 Publisher University of California Irvine (UCI) Release Date 2014 Language English

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