Merken

# Lecture 13. Hydrogen Atoms

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Erkannte Entitäten

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00:06

but were continuing on with the exposition about the hydrogen atoms when we last left our hero we had separated into angular and radio equations and we had made a substitution you is equal to our times are and then we'd shown that if we make the substitution would get a much simpler equation to solve now what we're going to do today we're going to talk about the radio functions and solutions of the time independent showing their equations but we will get these letters states that Rittenberg observed experimentally and his spectroscopic investigations and fit that famous formula the difference between the inverse squares let's then tackle our problems 1st by setting a little L is equal to 0 we know that a solution for the particle on a sphere and we can put that and if we put little Ellis equal to 0 and the term won over our square is out of our hair make gives us a much easier equation is solved my far squared over to new times the 2nd derivative of view with respect are minus eastward over 4 by epsilon are and you is equal to the what we want figure out then is what is you and what is he for this equation and now we've got a little bit of experience with Icahn value equations like this we've got derivatives of things and then we get a copy of the thing and every time it seems that the exponential function comes in when we've got stuff like this furthermore we can make an argument that look the way function has to go away 1 little or against big because the chance that the electron is going to be very very far away From the protons In a hydrogen atoms In the ground statelets say has got to be very small and in particular it has to go away when AZ infinity so we can't possibly have a polynomial and because 1 gets bigger upon only only the blows up

02:38

a blows down and it's certainly never goes away and so we're going to use an exponential function again but we can't just use an exponential function because we have this other thing 1 over Archer and we it's the 2 of them together not just the 2nd derivative like the particle in the box because the 2 of them together that have to yield a copy of the the function that we put into this the handle we're going to turn out let's try I'm an exponential but let's just try the simplest fix we could make instead of just using a regular exponential which we can see straight off work let's use are times the to the miners are upon our not where are not it's some constant with dimensions of length that's yet to be determined we should put constant also in the 1st term but further for the time being let's ignore the animal will put the cost the exponential because we know we don't take the exponential of something with you if we take the 1st derivative of you With respect are what we get is you over or minus you over or not after we do a little bit of algebra that looks very promising because now we've got something with 1 over are in and that's just what we wanna have if we take the 2nd derivative which we're gonna need to get the kinetic energy we get minus 2 over or not times you over plus you over on the square the question is can we take this and make it work in the item value equation and the answer is we can because conveniently enough the miners to are enough as a minus a spots where so therefore that's that term is now positive terms something over are and the potential is a negative terms something again over are both of our times you and so I can get them to cancel out if I make the 2 things the same and what that means is that a spot square over a new times are not has to be equal to the squared over for pie epsilon not that makes the 2 terms in 1 over are the same and that if I sold it for are not blessed me figure out what on this constant this length distance and it turns out that is for Absalon times a bar square over minutiae square epsilon not so well known constant age squared I know I have a formula I know the masses of the proton and electron b the electric charge 1 . 6 times that mice 19 who loans that got everything and I've got this this magic number now are not which it has something to do with the length scale this problem now that I've canceled out those 2 terms I have another term last and that termed whatever it is it has to be managed and the term that I have left is my estate buyers squared over to you are not square that term is the energy Oregon and if I saw them for the energy and then the addition putty in the solution Irish cancer or not in terms of the fundamental constants like the following equation for the energy energy E is equal to minus New Times eagled the 4th power that's the charge on the electron not the natural law divided by 32 price Absalon knots with age bias the energies negative because as I remarked on the last time our energy 0 0 is with the particle separated at infinity and there are more stable than them when they're the hydrogen and it would take energy to separate there are lots of constants in here but let's go put the man as a practice unless verified that you harness that had the right the idea 100 about 125 years ago and by comparing the constant we get through this lost value of the energy with what he got for the source of the this is practice problem 16 verify that our formula has for the energy has units of energy yeah that's 1 good thing to do because a lotta Constance here that doesn't units of energy you've made an algebraic mistake anybody can make an algebraic mistake people with experience catch the mistake before they speak about the publishers or otherwise revealing images quietly fixes and then let's convert this energy assuming it is energy and I have made mistakes and I was converted to waste numbers and let's see how it it compares with the 109 if you look at the units on Absalon not it can be frustrating you may find very odd units "quotation mark something in ferrets and men I gotta go look up a fairer and so forth and so that isn't so useful the way I remembered the units on epsilon artists just to remember the cool long Forest Law the force between 2 charged particles is the charge of 1 times the charge of the other divided by 4 pirates along our square and I know what forces that's Newton's and I know what charges that schools and I know what artist that's meters and so far it's all epsilon not the units of Absalon not 1 myself for it in terms of the others just looking at the unit's his cool on squared per Newton per meter squared and with this in hamlets but this into the formula for the energy -minus new readers for divided by 32 points where epsilon non-strategic possible it would put the dimensional and analysis we get telegrams cool almost to the fore cool loans to the minus 2 squared Newton meters squared meter square and jewels to the minus 2 2nd the minus to and so what we end up with and is a kg meters squared per 2nd squared conceptual so that's good we did make a mistake there what about the numerical values well let's figure out the reduced massive we can figure out what view it as a massive proton breast masses protons times rest mass of the electron divided by the of Proton plus the mass of the electron put the numbers to a lot of digits if I see something

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quoted to allow the digits I tend to enter it into my calculator with a lot of digits people kill themselves and work long hours to get these numbers to high accuracy and family the United be able to compare experimental results if you start rounding things off to 2 or 3 digits when we started out we have to keep them all and man and you may have to be careful how do the calculation you then turns out to be 9 . 1 0 0 4 4 3 times standard minus 31 kilograms In other words new is very close to just what the mass of the electronic and as the nucleus is heavier than we can kind of forget about new because as is nears we like is pretty much equal to the mass of the electron the other things not moving the other thing is not being told as the electron goes around because it's so light if we thought that were going around like a classical particles I'm OK so we don't want include any rounding on any of the other costs and so we have to write things out in gory detail on to his many digits as we can find and if we do that put all the numbers 9 . 1 0 4 4 3 1 . 6 0 2 1 7 2 3 8 . 8 5 4 1 8 7 1 7 and clients constant and work it out We get minus 2 . 1 7 8 6 0 0 times 10 to the minus 18 jewels which is a very small amount of energy but remember we're talking about one-half of course it's a small but what is it anyway numbers so well this is why we use wave numbers instead of jewels is that it's much better so to convert await numbers we just take that number in jewels and we divided by age C but we see in centimeters person always used at the time of the attack and doing that putting in the numbers again has to the correct position I find that are so they should be 109 thousand 673 and to at least 5 digits that's exactly what is observed and so could be that I made a typo in 1 of these constants why didn't I get it exactly right on the last engines and slight error but I work for it could be that somebody else who wrote down 1 of these numbers made a slight area there's some disagreement or could be the made slight but anyway you never get 5 digits in life on almost anything new measures so the fact that we get that to 5 digits is fairly impressive but keep in mind that quantum mechanics can make very accurate and precise predictions and if you want to verify if it's correct keep all the the precision on the fundamental constants don't start a calculation by rounding something off doesn't take that long to enter just do it the boiler radius is given the symbol a not and it's a special case of R value of on on which we got by putting an end to the differential equations and is just the special case for a nucleus that so massive that the reduced mass is equal to the mass of the electron because once once the nucleus gets heavier than a proton that's pretty much all she wrote on them and they're not Dennis for prior Epsilon beach parts were over the massive electron guns to charge of electrons square and that's about 52 . 9 Pico meters for about half an Inkster in the old system of units How could tell Excuse me even for a hydrogen you know the difference is only the 3rd so some books just use a not without even mentioning the reduced mass at all the problem with mentioning the reduced masses you have to explain the separation of the center mask ordinances that you're assuming that the nucleus is fixed on a lot of other things and if you've only got a certain amount of time to to cover the material and you'd like to draw the shapes the way functions you just start as if there were only 1 particle involved the electrons and you you you go forward are noblest way where are they got is just an exponential because keep in mind that we picked for you hard times are but you is art and they got so therefore are itself is just an exponential because the rest of us it's an exponential that has a cost but 0 and then falls away and I myself and that corresponds to the ground state energy and equals 1 In red birds formula that's the lowest energy that we can have for the hydrogen have and the experimentally as I mentioned in the last lecture this difference between 1 and squared suggests to us that the energies themselves there are won over and squares so it's minus 1 and minus a 4th minus 9 so on and then they start getting very very very close together and send it the and the proper solution for all end it is something you may do in a more advanced courses but I can explain qualitatively at least Hollywood Hollywood does approach them the standard method is to just assume an answer and put it in and how clever you are unassuming what kind of thing you can put on is how long you're going to spend solving the differential equation in this case you guessed that the radio functions they got along is some power series with some coefficients times are over or not to the king times an exponential we take some polynomial just like what we observed in the harmonic oscillator readers polynomial times e to the miners are upon us and then if we say while the 2nd derivative has to equal this and that we get the relationship between the coefficients a sub K and that relationship gives us some equations to solve and if we can solve that those that we can figure out what those numbers KKR R and then by magic we had the solutions this is the most general form because although it looks very impressive with Kate burying an integer powers of up to infinity and principle but in fact we could have some funny equation and some other context where the powers a half integer instead of integer the 1 have 3 have the square root of something times of Poland or something even more with negative

18:03

power but we have included here and there some powerful arguments as to why you would want to include those things that but you have to keep in mind that even a power series like this may not be the most general and then to try if it happens not to work you don't have to give up the contract and in fact when we do that all that we find we get exactly the same equations but now in the denominator we and square 4 for the energy and so does go like 1 over a square and enters a quantum number it has had has to be an integer and whenever I was equal to 0 the wave function is spherically symmetric and there are radial nodes in the wave function except for the case and equals 1 which is just an exponential for ankles to there are some radial knows a radial notices sphere where there is no chance of having the electron because the way function changes sign on 1 side is positive on the other side is negative it has to go in between 0 and when you square gets very small that so the probability of being in that Shell is very small and the show was all around the atom which seems kind of mysterious because you have a chance good chance of being inside there and a pretty good chance of being outside the region no chance of being being right there and that's because the electron is a standing waves and and waves have different properties than like a PDE rattling around or whatever the picture my head we know that you have to have so if 1 of them is the bottom was an exponential which we've got In the next once a polynomial we know that it has to change sign somewhere because when we integrate the end equals too times equals 1 we take that complex conjugate but we know if we integrate those that is and is equal to if they're the same year-ago should be 1 that means it's normalize the chance of finding electron somewhere at some value of art is equal 2 1 100 per cent but it's a different 1 then we have to steer away from her mission operators that operator with real life and values of the undervalued and different than the item functions are orthogonal and he said nothing to do with each other and that means that in a goal has to be 0 if 1 of them is always positive there's no way the other 1 can always be positive and when you multiply it at the answer come out to 0 so the other ones has to have nodes in exactly the same way that further Hermy polynomials for the harmonic oscillator they had happened otherwise it wouldn't work out and so on the total number of nodes it is an minus 1 and if that was equal to 0 there are all these notes are really knows they're all spheres where the wave function ventures but if is L is equal to 10 minus 1 which is the highest value fell that's allowed then the notes are all angular nose and what that means is not what most people think when they hear angular known angular notes means there an angle where the wave function vanishes and what that really means is considered the suppose the wave function vanishes when data is equal to 90 that means when this is equal to 90 the wave function is vanished nothing else matters that means vanishes on a plane so angular node is a plane where there is no electron density radial note is a sphere the service of the sphere where there's no electron density those of the 2 kinds of so that we can have a Friday region and in between the 2 extremes were there all angular nose or already all of you can mixture of each you can have a place where there's no electron density it could be they appear that down here and then intersecting that could be a sphere where this thing starts positive and then changes asylum becomes negative outside the the sphere and vice-versa and then in effect would be a picture of a so-called three-peat orbital which will encounter them later historically when we we adopt for historical reasons names that don't seem to make too much sense but they do make sense if he had filmed back in the old days Indiana Cameron you you're developing the film and you saw these lines on the film where you were analyzing the frequency of the light by how far prison with standard some lines were shot they were called and they were given the abbreviation that's for sure and then some lines were the main line she saw so they were called the principal line and given the abbreviation P and then there were some other ones which were colleges use and they were given the symbol D and then there was another set that had lots aligns closely spaced and seemed to have what they called fine structure and they were given the symbol that and we start with now and would now we know but these correspondent different values of that the orbital angular momentum if the orbital angular momentum now is 0 we called it's in accordance with the sharp line if the orbital angular momentum is what we call a P if it's too we call it the if it's 3 we call back and then if it goes higher than that we would be G and agents of for Adams that stable in the ground state the periodic table we don't have to go higher than that because we ran out of gas for Adams around 92 with uranium because the nucleus itself becomes unstable and so although we could have all these nice electron orbitals filling up while we try to to jam so many protons and we try to piece them together with enough neutrons to hold things together it's not stable enough and so it just falls apart on its own and so radioactive elements have long since decay if they were around when the Earth was 1st created and if we labor solutions by end and now we can write them down and the latter's often as a letter so we may write 1 yes rather than 1 0 0 for the wave functions we saw before on the particle on a range that that party had eaten the eye and fly and

25:46

that's kind of difficult plot because it has a real part of an imaginary part and they all have the same energy because the energy only depends on an doesn't depend on or am I so well foreign observers and therefore we can just take combinations of different values of em such that we get something that's nice plot and that therefore if we have the 2 the I am Friday and we combine it with you to the minor and 5 if we Adam up we get to and if we subtract them we get up

26:30

to I signed fine and then we just normalized to co-sign fine signed 5 and the week those we comply easily because those are real things and so those are the ones we tend to use their not proper and functions of of LCD anymore but they're much easier to plot because they're standing waves from traveling waves that go around like a corkscrew here the 1st few solutions that for the hydrogen on site 1 as we have normalize 1 over square apartheid 1 over a nod to the 3 times the exponential just goes away site to West is the same thing the normalization process a little difference because of the other terms we have analysts have to make sure we have 100 per cent probability of finding an electron somewhere and now we have to minors or upon enough so that we can see the note because when you are upon a nod is equal to 2 it's 0 and then we have the to the miners are over to a knot instead of a not so we get to up there 2 a we write the the same thing and now rather than the term to minors are over a knot which has radial no we have an additional terms "quotation mark and co-signed data as an angular nose because 1 thing is equal to 90 "quotation mark the changes sign so this is it's plus on 1 side and minus on the other side the 2 PX into PY functions I'm not because as I mentioned on actual like functions but they're easier to apply and they are signed data and then co-signed 5 4 2 PM and signed dated and signed 5 4 2 3 1 and so we can make lots of these we can say with what is that we can draw contour of 90 per cent the probability 90 per cent of the electronics inside that and we can draw the shapes and those are the kinds of shit juicy drama books especially freshman chemistry book the 2 P 2 P 1 to P way functions are all the agenda they all have the same energy but they must have some symmetry relationship With respect to each other the total number of the general solutions for level and is an squared not taking into account the electron spin which could be the upper down and that would give us another so would be to and squared so that that is an indication that a single hydrogen atom has a very high degree of cemetery because it has a high degree of the generously so that means that the it's very nice system very symmetrical and that's accords 1 reason why we can't solve but I've ridden the wave function for site 3 yes it's a quadratic an are and then you the miners or over 3 and a half 3 PC has a co-signer data and now it's only a linear terms of steps down 1 radial known disappears and endangered animal appears and three-peat X again we end up with scientific Coast and through the wine we end up with something signed 5 and there are 5 solutions for the L equals 2 cases we had 1 and we had 3 we had 5 the toll is going to be 9 . 3 square that's how many we said we should have a three-piece 3 what's called 3 D Z square is big normalization constant to the miners are over 380 not again and then 3 co-signed Square Theater minus 1 we get a new polynomial invader nothing has stepped up on that because we have a L equals to we have to quantum of angular momentum and for I X C instead of "quotation mark squared data we have signed data co-signed data "quotation mark signed that's just exactly like it behaved before for the 2 and for the others follow the pattern for why we have signed it cos they signed 5 and then we have to evolve 1 owns squared minus y squared that science squared data co-signed to fight let's get to units of twist on the end of the magnetic ones and then the other 1 x y sites "quotation mark their signed to fight as I remark we just take these linear combinations because they're easier to plot not because we were thinking that those are the values of else eh In this next slide what I've done is I've plotted the wave function itself as a function of and normalized to get rid of the constant and they know and all they not cute and all that stuff going on to the street what you see is the kind of cell also sigh 1 has come swooping down and buy a new articles about by our own over a new articles to so twice the border radius is pretty small and by 4 it's very small by tenants essentially 0 that's as the 1 asked according to this picture of looking at things in it it disappears pretty quickly and then the 2 as there is a no-load got near to and then it becomes negative and then it just dies out and once again by 10 it's pretty small and by 3 as it as it comes down again and there comes back and then comes back through just a little bit between 1810 and then dies but you can see that most of the density if you apply this way seems to look like it's very near the nucleus it doesn't seem like there's so much difference between them and that's misleading because there actually is a lot of difference what we ought to ask is we ought to ask the following if I'm a distance are away anywhere since its spherically symmetric and I'm out there what's the chance that I'm going to find the electrons in that show and that's different because I have to wait and if I wanted the probability I have to square the wave function number 1 and number 2 tho I wanna get the probability in any shall have to multiply by our square and let's do that that's called the radial distribution function which every parent's role and if I do the role 1 answer I see that when Myers very small there's no hardly any chance probability there of finding it because the shell is too small so the yacht square is small and then it goes up to a maximum and right at the border radius is the most likely place of finding the electron in that Shell right there and then it dies out and bite-size they've not the chances pretty much 0 but now let's look at at 2 West so that little excursion at minus because it gets blown up by our square and the fact that we have anything out there because the show was so big the chance of observing the electron shell out there is actually much higher than in the interior but this is much more like what we expect we see there's little blip inside and then there's this big hot where the electron density can be there's 2 important things here the 1st is they look like shells because 1 the first one dies out that's where the 2nd 1 starts coming in and it gives us a

35:23

result that's important and the 2nd thing is that the 2 of electrons can penetrate there's a little

35:35

blip inside near to the nucleus and so the 2 us electrons can spend some time quite near the nucleus even inside where the 1 s electrons it could be they work both there together as they will be in some other atoms and that's an important feature as well because it means that the 2 US electron knows what the nuclear charges and like helium lithium in a way that the Tupi electrons doesn't have any chance of being at the nucleus cannot and in multilateral Adams will see that that changes the energy quite a bit then over 3 for the electron density for the 3 us I see there's a little blip nearby and then another little blips and then there is a big blow and it

36:26

gets big again right where the other 1 dies out so this is a picture that the ad is like and shelves like an onion it has shells and they can be occupied by electron that's that's a very important thank to to realize In spherical polar coordinates we can we can calculate the probability of being at a certain volume and Florida as an example problem let's just do this calculation for a hydrogen atoms in a wondrous electrons unless to following let's calculate the chance that the electron is within a sphere of radius knocked border regions so this is practiced problem 17 I just figure out the probability above what's the per cent chance the unified the electron nearer to the proton the border radius near equal well have we really know the way functions for sigh 1 we can square there's no complex numbers so we don't have to worry about the complex conjugate we square and we know what DVDs the volume for spherical polar coordinates are squared signed data Dee thing defiled and so were left doing an integral which I've written here the way things used to be written in books I have a triple integral From fight will 0 5 calls to pilots and bring them here from data equals 0 0 to the it's that part and from our equals 0 a non that's what I'm stopping but the wave function square defiance Sunday at the Theatre Arts where and did this dependence is pretty easy because the way functional independence on

38:31

R and decided that positive at the great so we can do that but it's it's much better I think to right the formula this way what I've done here is I have had the right fight 0 2 5 was to high because what I've done is I've put the deed something right next to the integral part the integral defile and then I just say 0 2 Part

39:00

it's understood that's referring to fight and I have been in grant and then I had the next 1 the data next Dr and then I don't think I can clean it up so it's a little bit nicer the Honourable Elsie iDecide I but you know I don't have to look back 1 because that 1 could be easier the inability whatever it is the whatever benefit in I'm integrating from 0 to piety to Pineland 0 I get too for the data I just have to integrate from 0 price signed dated the data that's minus co-signed data it's not a big deal and if I couldn't find 0 idea minus minus 1 and minus -minus 1 and careful and to therefore if I do those 2 integral and left with the following I get to high from the fight 2 from the theater and then I are square and I have this thing the wave function square interface square opining on the game minus 2 0 on the nite so I can clean things up I get following in a goal to do for times won over enough to the times integral from 0 to a knot of our square either minus 2 our upon a new deal as a one-dimensional integral and you can look this up and what I've done here as I've taken a screenshot From the roles . com and I've put in you have to put it in terms of that's not are not very smart that way but I don't put an X squared exponential function minus 2 eggs over a nod In click here and it gives me this format about whether this is the time to compute was a hundredth of a 2nd and it says it's minus 1 for the cannot be utilized to acts upon a knot times you may not square close to a Nymex to square so I just grabbed that and that's pretty easy if I accept that that's true enough I doubt it's true I can certainly take the derivative and see if it's true I don't know if there is a derivative stock ,comma maybe that's too easy and I put that in and I simplify it and put in the limits of 0 for acts or on parties and I put in the limits a knock on the top of the what I it is I get a knock you go over 4 -minus 5 a not you over 4 times each to where he is now the natural laws he not the electric charge and if I evaluate 1 minus 5 minus 2 I get the number . 3 2 3 excuse me 3 2 3 2 3 sometimes they not you for now I go back and collect everything together that I've got the ah I have I had 4 has 1 over a not too and then I get a knock you go over 4 great Everything goes away and the only thing that's left is this . 3 to 3 and so the answer is but the probabilities . 3 2 3 and that means but it's only 32 . 3 per cent likely that if we take a hydrogen atoms and we measure the position electron that's going to be within what we think of normally as the size Of the hydrogen as we usually think of a on something like the radius of the hydrogen atoms but it turns out that this much squishier than that it's only 32 per cent time Maryland and these arrests the timing of the electron is roaming around pretty far from home but kind of rather surprising so the interpretation is the electron is free to roam and can get pretty far away and that could be very important because when it gets far away it may have a chance of being picked up by by some of the positive nucleus may actually go somewhere and make a chemical reaction if you were held to tightly very close to the that would have it

43:33

that would have ramifications for reactivity and although the border radius is the radius where it's most likely in that show it's not actually so likely to be with American actually the outside that as well In higher orbitals the chance that the knew the electron can be far away From the ad it is very very high the saw 3 S 3 is not that high in energy in terms of quantum numbers it is actually when you look at -minus 9 versus minus 1 it's 90 per cent of the way toward disappearing into a

44:19

Proton hole and electron but I did have a very high chances the electrons very far away and so something can happen when you're very far away you can collide with some other things and you're gone and in fact you can specially prepared Adams in states called Red Bird states named after Johann stripper which both an L are very high what that means is that emphasizes the energies and Alice so that means there's a ton of angular momentum so if we imagined then spending it's very much like a classical object that because is spinning so fast and it's out some radius if we actually look at those if we prepare Adams very carefully and backing don't in anything we can make Adams there's because the fleet by doing that and they hang around for quite a long time until they collide with the other container something else happens it's very surprising that you can puff up anatomy but puffing up the electron to some very high level by some tricks with lasers and other means To get these kinds of reinstates and that they are that the electron is so far away but still barely stable course anything happens then he hadn't OK I'm going to stop talking about the hydrogen atoms there in terms of the structure of the wave function and the quantum numbers and the solutions we have if we have to use the solutions of course we just look amount and in the book we don't try to read arrived the whole thing but it's important least 1 time to see where things came from fully understand what the arguments were that led to the conclusion that were were stating as if the facts next time than what we're going to talk about is atomic spectroscopy we're going to talk about why we got these 1 over and square and the series and how we can tell what the when an animal ionized how we can analyze data from that wavelengths of light emitted from an and figure out if it's going to ionized and so on and if so what energy will it take to get established that the cost that's 1 of the fundamental things those quoted in the data in the periodic table is the ionization potential of every neutral and melodic chemist understand reactivity and other things by way ionization potential is high or whether it's it's pretty low and will pick it up there then the next time and start talking about atomic spectroscopy selection rules electric dipole allowed transitions and probably a little bit about terms symbols as well so that we can have a nice way to catalog all the transitions and understand which ones which a very nice shorthand for labeling them without too much ado will leave it there will pick it up next time with atomic expects costs where

00:00

Protonierung

Derivatisierung

Kryosphäre

Fülle <Speise>

Elektron <Legierung>

Chemische Formel

Nanopartikel

Vorlesung/Konferenz

Funktionelle Gruppe

Lösung

Atom

Lösung

02:37

Mineralbildung

Symptomatologie

Zellkern

Lösung

Konkrement <Innere Medizin>

Aktionspotenzial

Edelstein

Altern

Derivatisierung

CHARGE-Assoziation

Elektronegativität

Nanopartikel

Chemische Formel

Funktionelle Gruppe

Zunderbeständigkeit

Systemische Therapie <Pharmakologie>

Atom

Lösung

Entfestigung

Elektron <Legierung>

Symptomatologie

Potenz <Homöopathie>

Krebs <Medizin>

Zellkern

Querprofil

Quellgebiet

Rydberg-Zustand

Tellerseparator

Protonierung

Stickstofffixierung

CHARGE-Assoziation

Chemische Formel

Mannose

Bohrium

Advanced glycosylation end products

18:02

Single electron transfer

Zellkern

Ordnungszahl

Orbital

Lösung

Atom

Chemische Struktur

Kryosphäre

Sense

Nanopartikel

Vorlesung/Konferenz

Operon

f-Element

Uranerz

Funktionelle Gruppe

Wasserwelle

Atom

Lösung

Konjugate

Kryosphäre

Elektron <Legierung>

Symptomatologie

Potenz <Homöopathie>

Quellgebiet

Zuchtziel

Protonierung

Radioaktiver Stoff

Mischen

Emissionsspektrum

Alignment <Biochemie>

Zetapotenzial

Advanced glycosylation end products

Chemisches Element

25:45

Mineralbildung

Chemische Forschung

Laichgewässer

Zellkern

Ordnungszahl

Lösung

Teststreifen

Simulation <Medizin>

Elektron <Legierung>

Alkoholgehalt

Vorlesung/Konferenz

Wasserwelle

Funktionelle Gruppe

Systemische Therapie <Pharmakologie>

Atom

Lösung

Aktives Zentrum

Zelle

Fülle <Speise>

Elektron <Legierung>

Wasserstand

Mähdrescher

Gangart <Erzlagerstätte>

Stoffdichte

Ausgangsgestein

Erdrutsch

Toll-like-Rezeptoren

Emissionsspektrum

Adamantan

Chemischer Prozess

35:23

Zellkern

Elektron <Legierung>

Elektron <Legierung>

Lithium

Helium

Lactitol

Ordnungszahl

36:26

Konjugate

Kryosphäre

Elektron <Legierung>

Atomsonde

Nahtoderfahrung

Konkrement <Innere Medizin>

Schelfeis

Protonierung

Protonenpumpenhemmer

Chemische Formel

Elektron <Legierung>

Chemische Formel

Massendichte

Funktionelle Gruppe

39:00

Humangenom-Projekt

Click-Chemie

d-Orbital

Grenzfläche

Chemische Reaktion

Elektron <Legierung>

Zellkern

Reaktivität

Klinischer Tod

Orbital

Stockfisch

Derivatisierung

Kryosphäre

Elektron <Legierung>

Sammler <Technik>

Vorlesung/Konferenz

Massendichte

Bohrium

44:18

Blätterteig

d-Orbital

Wasserstand

Elektron <Legierung>

Symptomatologie

Reaktivität

Adenosin

Dipol <1,3->

Ordnungszahl

Lösung

Chemische Struktur

Kryosphäre

Übergangsmetall

Elektron <Legierung>

Wildbach

Pharmazie

Spektralanalyse

Bisacodyl

Massendichte

f-Element

Ionisationsenergie

Containment <Gentechnologie>

### Metadaten

#### Formale Metadaten

Titel | Lecture 13. Hydrogen Atoms |

Untertitel | Radial Functions and Solutions of the Time-Independent Schröinger Equation |

Alternativer Titel | Lecture 13. Quantum Principles: Hydrogen Atoms |

Serientitel | Chemistry 131A: Quantum Principles |

Teil | 13 |

Anzahl der Teile | 28 |

Autor | Shaka, Athan J. |

Lizenz |
CC-Namensnennung - Weitergabe unter gleichen Bedingungen 4.0 International: Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen und das Werk bzw. diesen Inhalt auch in veränderter Form nur unter den Bedingungen dieser Lizenz weitergeben. |

DOI | 10.5446/18891 |

Herausgeber | University of California Irvine (UCI) |

Erscheinungsjahr | 2014 |

Sprache | Englisch |

#### Inhaltliche Metadaten

Fachgebiet | Chemie |

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: 0:00:57 The Radial Functions 0:07:12 The Rydberg Constant 0:13:39 The Bohr Radius 0:22:54 Notation 0:26:55 The Wavefunctions 0:31:59 Radial Nodes 0:36:44 Integrating Wavefunctions |