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Lecture 12. Spin, the Vector Model and Hydrogen Atoms


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welcome back to chemistry 131 a today we're going to talk about spending the sector model and begin to talk about hydrogen atoms a subject that will continue on for a couple of lecture recalled that when Stern and the elected their measurement there were 2 bands that were observed rather than a continuum of silver atoms distributed and that was interpreted to mean that there were 2 possible states for the angular momentum other magnetic moment the intrinsic magnetic moment of the electron and that would correspond to answer best equals 2 plus one-half or minus 1 half and the interesting thing is that it was a half rather than integer which made it seem fairly mysterious at the time In the vector model of angular momentum we I visualize the angular momentum vector lying somewhere on ,comma and fizzy component we imagine that we measured and that has a definite value and we've measured the total angular momentum that also has a definite value but the X and Y values are and determinant for reasons that will see In a minute for an electron then they're just too such orientations of this particular cones and we tend to just call them up and down and recall that we don't actually physically think anything is spinning or rotating more there's a little planetary model inside the electron we have no evidence at all for anything like that this is just an intrinsic property that there is a magnetic moment and it behaves as if there were an angular momentum and that the 2 projections as himself sequels plus reminds 1 we can do spectroscopy we can do magnetic resonance we can make transitions between these 2 magnetic states and a lot of information about the molecules and photosynthetic systems and various other kinds of things have been cleaned by actually seeing as there is an unpaired electrons what the energy differences between the 2 magnetic states and actually getting a spectrum of possible difference and seeing how that spectrum may change if we radiate the sample with light or do other things to influence the molecular structure both arresting one-half we've got a picture like this Of these 2 columns 1 call 1 Kona down these are the 2 possibilities and there is a definite value of the sea component of the angular momentum age far over 2 and the other possibilities minus a bar over to 4 the X and Y components so there is determined and that's why we just rather ,comma because that's meant to show there was the X and Y components could be anywhere on this call will see 1 2nd 4 a higher angular momentum we would have more Council will see that SEC what about the usual conditions that the wave function whatever it is the single valued and have at least a 2nd derivatives if we put and instead of EDI and 5 imagine some kind of thing there with them over to them the problem is that the changes signs but keep in mind that Spain is different because when we arrived there the eye and fight we were actually had physical particle on a physical ring and we got those functions from the boundary conditions there but here we have an observation of a magnetic moment but we don't necessarily know that it corresponds to anything like that there is no spatial component to it exists just as the magnetic moment of the electron but there's nothing moving around in space that we can answer and the US spin part can actually make some calculations quite difficult because for example if you have a molecule with an unpaired electrons then it may be that whether standards upper down changes the spatial electron density quiet because there are a lot of other magnetic particles around the molecule nuclei other electrons and so on and it may be that if you spin has 1 orientation the magnetic moment that hangs out more over here and if the spin has opposite orientation that it ends up in a different part of the molecule and that means that if you're trying to keep track of chemical reactions were electrons can become 1 pair of various things can happen sometimes the calculation gets much more complicated than you would like because you have to know something about the spinner average over the figure out what's going on the spin one-half if you want to think about it is rather like a Moebius strip in that if I put a twist in a piece of paper then rather the coming around like a ring and going the same way if I could just as the twists and taped together when I come round I'm actually pointing inside rather than outside and when I come around again I'm going outside so the wave function repeats when you rotate by 4 that so-called spent her behavior and usually it's extremely hard to observe but if you have a reference state and then you have a state that you're rotating with respect to the reference and you do this in magnetic resonance experiments you can actually see that when you apply also that you actually see the thing change signed if you go around by 2 Pike and if you go around but for part it comes back to the sea sign again here is a picture for the spatial part here are the 5 components of an SUV 4 an L equals to state which will see is called the orbital and again the columns have to match and the length of pasta also match because we imagine we measured at all so we know well well-timed selfless 1 and then we measured the sea components so we know that and get this series of columns and again the LX all-white components are completely and determine but get these pretty pictures and this is the so-called Vectra model for angular momentum will see an atomic spectroscopy in a couple of lectures that keeping track of the angular momentum of the electron depending on what the value of ballot hands and keeping track of the spin and depending whether tougher down is very important for keeping track of where the spectral lines will appear and to assign the spectrum to assign spectrum means that we know the energy level where the electron started and then it falls down to a lower energy level and emits light of a certain frequency that comes out someplace and we know that the energy levels and we know as many quantum numbers about each energy level as it's possible to know if we know that when we say that we assigned the spectrum if we can match up everything and then we can infer a lot about the structure of the DNA by where these lines actually happened to be like the sodium D lines for example that give the characteristic yellow color of the sodium vapor lights why do we have this uncertainty well it turns out that this is another manifestation of the uncertainty principle at this point rather than just talking about Delta P Delta X we can make a connection with a much more general idea namely if we're going to make a measurement of 1 thing another thing then
if we make the measurements in reverse order the question is whether we get the same result or not if we get the same result by making the measurements the other way around that we imagine that these measurements are compatible and it's OK if on the other hand we get a different result like we saw with the coin worth we wanted to see if it was heads and tails or we wanted to see how thick it was that we ended up getting different results at random for whether it was Heads details after we measured the thickness then ,comma they're incompatible and this incompatibility is actually coded in the operators themselves so we don't have to guess about we have a mathematical way tantalize with the operators of doing all we can figure out but it turns out that the square of the angular momentum at all times 0 plus 1 or as times as plus 1 embassy component or any other components but 1 were picking a component we the buck convention they're compatible they can both be determined no problem at all no problem with the calling on him but if we tried to an addition determined in addition to Elsie and elsewhere if we go further and say Well I'd like to also know know why as well things get fouled up and what happens is that we measure 1 of those than LCD gets changed at random and if we go back a measure Elsie the other 1 gets changed at random and no matter how many times we go back and forth we can't get consistent result because the or the result we get depends on the order that we measured in and furthermore it's Randall doesn't pan like a flip-flopper or anything like that something orderly that we can predict it's just a random value and that he is whether the operators commute and the commentator really as you put 1 operator on 1 side and the other 1 behind it and then you do it the reverse way put this 1 on this side and this 1 behind it and put the ball on the way function and then is attractive and yet USA Track them when it comes to 0 then they are compatible and you can measure them both as accurately as you want and if they do not and they are incompatible and that means that making 1 measurement will destroy some of the information that you gleaned from the previous matches let's just take our operators for positioned there and momentum position remember the ex-head operator was just multiply the wave function by X and the P had X operator was to take the derivative minus ii each Bach times the derivative with respect suppose we want figure out the commentator then we write this square brackets and we X ,comma key and what that means is that means take X PX -minus Pierre X so let's do that the 1st term is and then on the right-hand side minus ii H bar D by IDX that he had minus the opposite order which is mine side body by DX of and what we do that is we put our way functions on the far right Of this we don inserted where it might seem to go in next to the derivatives past the access we put it on the far right hand side of these operator equations always remember that don't start certain things in the middle he to the right because the way operators work is left to right by convention let's apply this commentator to some way function sigh of experts completely arbitrary we don't care what it is as long as it follows the rules for the the continuous and have derivatives and so on the 1st term then it's quite easy because it's times minus Irish times a derivative of sigh with respect X whatever that may be the 2nd term is now classified as because of minus minus 4 times a derivative of the product some and so that means I have to take the derivatives of X then times side a plus the derivative of sun and facts and if I do that then I find that 2 of the terms vanish the minus side access these side cancels with a plus side by architects Dietz idea but I end up with that 3rd term because of the rules for taking the derivative which is age Bok side and so the commentator ex-head peace had X applied to the wave function returns a number which is real but it has died at his age H-bomb some therefore since is arbitrary what we say is that the commutator this this relation has nothing to do with size so at the end we eye out and we have an operator relations the operator head ,comma he had X is equal to my age bar and key is it's not 0 since it's not 0 we conclude that we cannot measure X position and X momentum simultaneously the arbitrary precision I and I've written this relationship here on slide 312 this is just an operator relationship you can think of this as I age so 1 operator 1 where the 1 operator just multiplies the side by the number 1 and any time in fact the commentator of 2 her operators is non-zero doesn't matter so much well what it is and to give you a hint about how it's going to behave but the key thing is this 0 or is it not 0 it is not 0 they are not compatible if it is 0 so they are but in fact we can take the 3 components of angular momentum L hats that's remembered that these components adjust are crossed hh and that where we take effect across product so it can take Al Habsi as the angular momentum operator for the sea component that's exactly what -minus why he acts and telexes why have PC minus the what and I know why is this the hair PX minus access to sea if if we take these because we now know the accident PX don't commute do y PX commute yes sir they do because there's no 1 I take the derivative with respect to y of X times the wave function it doesn't I don't need use the product rule because these are partial derivatives and so if there's no there but I don't bother taking the product rule X is treated as a constant so I can measure the momentum 1 way and the position another way In principle as accurately as I like I just can't measure them both and I certainly can't measure all 3 to localize the particle plus known its trajectory if you work this out you can verify on your own that in fact the commutation relations between angular momentum are a little bit more interesting the commutation between position momentum is just a number of imaginary number but still just a number in fact if you take the commentator Alex with all y what to get as I age part-time sells the we not only get a
number back but you get another operator and if you take the commentator of with L acts you get IgE far L Y and if you take a commentator with you get fired for L X and when you look at all these things are in cycling permutations of you start with the 3 operators like this you're allowed to all this 1 here put this 1 here and move back there and you can do it again and you always get the same result as color cycling permutation and it's very interesting that these cycling permutations looks very much like what a rotation would look like we're going around like this and every time we take commentator we go around to the other components and in fact there is a very deep connection between the structure of the commentators and the fact that these are in fact referring to a rotation of something that they are angular therefore for the span I'm angular momentum we don't have we don't have our crossed the we just have this behavior but now all you have to say as we propose that there is such an operator's suspend angular momentum we know that because we have this magnetic moment we know it has a value of 1 half and for the for spending and that the 2 orientations and we just propose that the 3 operators we call them as sex why and St and that they follow the same exact rules they follow the same exact rules as the orbital angular momentum and that in fact but it's not really the derivatives and so forth but there are important but what's more important is whether you follow this particular rule or not if you do you're going to behave like an angular momentum and that's kind of an interesting way to formulate things and in fact as I remark here in more advanced courses that you take on quantum mechanics we just simply propose that there exist there are operators and they follow certain commutation relations and we don't need to specified what they are in concrete terms we don't have to say Well this operators a derivative with respect to access or anything like that we just say said he had is equal to far anything that follows that is going to be a and like position moment we don't have to write derivatives and various things like that why would we want to do that well there is a kind of a minimum of minimalist approach To the theory the last things you have to actually assume that the detailed than the stronger euro theory has because it doesn't depend on whether those assumptions are correct if you didn't make them for example implicit in taking the derivative is sort of the idea that space is a continuous I think that there is such a thing as dioxin there's there's an infinitesimal unit of space were taking the derivative with respect to women using these nice stylized functions to represent reality but what if it isn't what if spaces digital or sort of like a billboard when you get up close and you see little dots and only when you get back away doesn't look like a continuous spectra photograph well then you get to reformulate year-old periods that depended on all that kind of thing where if you just say Well I've got the commentators and that's it you don't have to do anything different because you come up with something else that has the same commentator and everything else in the theory still works whereas if everything in the theory depends on this linchpin while there is a derivative and all this stuff that isn't quite true then you're in trouble you have to start over OK we're going to talk in a couple lectures about atomic spectroscopy and in all kinds of spectroscopy and rotational spectroscopy and vibrational spectroscopy and atomic spectroscopy you run into this unit of energy that seems a bit off it's called the wave number and it has a very strange unit because the image is centimeters to the minus 1 and that doesn't seem like a unit of energy of all and then why isn't centimeters and so I want to take a little bit of an aside here to go into why we use this year the main reason is that wave number is about the right size there is a reason why we have measurements like the foot and stuff like that because it's about the right size to measure things that we encounter in real life and what we're writing down energies were writing down transitions and so forth if we express the energy and wave numbers we get numbers like 1 10 and 100 maybe a thousand but we don't get numbers like 10 to the minus 34 or tender plus 20 and chemist like big numbers like that because it makes it harder to to understand things and talk about it and relate them it's always a little bit more daunting when we've got huge and tiny numbers and words were dealing with 4 out of 4 atomic and molecular spectroscopy than the relationship between the energy and the way number is he is equal to age New that we knew from the photon and we can write that in terms of the speed of light because see the speed of light is a wavelength times the frequency you can see that because if you've got a certain way of life and the frequency changes and that's how far removed so that's wavelength times frequency is in fact the velocity the wave which relied is always see in fact go for the the frequency knew we substitute seal on land and then we set 1 over land that is equal to new bar and we say that the energy it stage times new bar and new part is the way the number that's what we're quoting 1 1 would do that and once wave number or 1 centimeter to the minus 1 corresponds to about 30 gigahertz and the reason why is that 3 times standard the tender is how fast like goes in centimeters per 2nd so that's the conversion that between wave numbers and hurts and it's much easier to visualize 1 wave number 1 unit as a wave number than 38 gigahertz of 3 times 10 to the 10 per 2nd as a frequency they they're at the same thing and we can measure the thermal energy and we can see that the wave number is makes sense in that regard but as you may have learned in freshman chemistry the random thermal energy that's around and can can be accessed by the system at some temperature T and colony of course always Kelvin is Katie that's about the rent that's about the range bowls Mann's constant let's figure out how many wave numbers this it would take a tea estate 25 Celsius to 98 . 1 5 killed put it all in Carson's K 1 . 3 8 times as much as 23 jewels calendar to 98 . 1 5 Calvin H. tenements 34 tools and seconds and then all we have to do is put in centimeters per 2nd because we see that the jewels go away the per 2nd goes away the Calvin's go away and so if we just put C and centimeters upper 2nd rather than meters per 2nd we get the answer and inverse enemy and what we find them random thermal motion is about 207 wave numbers therefore if we've got energy levels space less than 207 wave numbers then just things knocking around cannot stuff up into these higher levels because there's plenty of energy around to do that you
can think of the 207 as if there's lots of hundred-dollar bills on the ground just lying around randomly everywhere then is pretty easy to buy a cup of coffee if that's the way it is if it's very cold there is no energy there's no money anywhere there's a couple of cents there's no way you're going to buy a a cup of coffee so that you can make any kind of molecular transition in that case electronic excitation of Adams it is usually much higher then this number 207 . 2 wave numbers and that makes sense because materials only global it's a red-hot we know of the glowing the emitting light that electrons are making transitions in the material but that only happens if it's red-hot if it's actually very very hot it's not going to happen for any kind of metal or material like that at to hunt at 25 Celsius they don't emit light and so that explains why the numbers we're going to get for electronic transitions and Adams are very very much higher numbers than that and that also explains for example why infrared light to he succeed as long as you don't get too much of it because the photons are low but doesn't do anything to chemical bonds and neither does the radio waves microwaves from your cellphone regardless of what people say To the contrary don't know anything about molecular structure or radiation whereas UV light if we figure out how many wave numbers that has that's different that has the potential to break some bonds and that's why we can use it to sterilize water when we go camping because if we stick and hence LED YOU VE source it breaks bonds in all the bacteria the swimming around in the water across the very my new demands that the key is when you drink it are they alive and then they started doubling quadrupling and so forth and make you very sick or are they there but they're dead because the bonds have been broken by UV light same thing with your skin you get too much you get sunburned very bad but now an an atomic spectroscopy it was experiment 1st and then theory next in fact most of the time it's experiment 1st and that theory comes in after 1 once you know the answer then you can appear to be quite smart it's only very rarely say with observation of Mercury but prediction by special relativity and in a few cases like that where the prediction was made and then the experiment was done and it actually agreed with the prediction because the theory was so deep and in this case and dedicated spectroscopy Rick Berke a Swedish man found that In 1892 good lines the emission lines from a hydrogen are so what's hydrogen are well I can take hydrogen gas that's a diatonic and I can't quite put a lightning strike through it but there's high-voltage and 2 things will happen the bond will break I got so much energy there cascade of electrons slamming into knocking things up and then the isolated hydrogen atoms then will have electrons in very high orbitals a random number and depending how they guy make the arc and then they started emitting light and then I recorded all the like the that and I have a look at it and quite how he did find this is a mystery but he was must have been very very good with numbers and some people are very good with things like that and others are not so strong and he looked at where these lines work and he worked out that they follow this relationship that the the ways number of the the line appeared to be a constant times the difference in 1 over and 2 square where and to is an integer minus 1 over and once worked for example I think it could be too squared it for 1 square it's 1 so you get that 1 and then there's 3 and 2 and so forth in the solid and this number from are sub the Ripper constant referred to hydrogen I was this number 109 thousand 677 where numbers quiet followed that with this I mystery but it was interesting that followed them and what was even more interesting especially now quantise Asian quantum mechanics not continuum mechanics is that these numbers were integers that was a crucial observation because now this is a simple thing hydrogen 1 proton and 1 electron and it's giving you this clue that there are numbers and their integers
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


Formal Metadata

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

Content Metadata

Subject Area Chemistry
Abstract 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: 0:00:22 Spin 0:02:56 The Vector Model 0:11:06 The Commutator 0:22:25 Wavenumbers 0:29:45 Atomic Spectroscopy 0:34:34 The Hamiltonian Wavefunction 0:42:52 Separation of Variables

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