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Advanced Quantum Mechanics | Lecture 3

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a U.S. go we talk last time about angular momentum operators and I started to show you how the algebra angular momentum operators works how multiple plates with different angular momentum have certain numbers estates and the states for me the multiplex welcome back to a momentous but let's just got more generally about what I angular momentum is about after a good example is an object the particle movies and central force field center pardon was around the central force field by it has a radial position and an angular position the angular position really needs to angles and azimuthal and a polar angle of 3 dimensional system and angular momentum is conserve angular momentum as a vector and the angular momentum always it is in the plane perpendicular so to emotional plane perpendicular to the orbit so there's the angular momentum the orbit your new planes perpendicular to it because the angular momentum is conserved as a vector B orbital plane is also concerned the orbital plane stays the same but costly radial position was around the circle notice that pig-a-back moves around in some audit that's classical that's classical quantum mechanically the particle is described by function be angular momentum operator has to do with angular dependents of the year wave function in general wave function and a situation like this would be a function of position of chorus so as a function of position could be a function of X Y and Z but you could also write it as a function of the radial coordinate the radial distance and 2 angles polar new triangle we talk a little bit the 1st time may be the 1st time in the last 100 years we talk a little bit about angular momentum in 2 dimensions tobacco that for a moment and everything was towed then the angular momentum would just b the generator rotations about the about the perpendicular axis here and it's structure as a quantum mechanical operator would just be angular momentum is equal minus body kind derivative with respect to the angle at stake out a lot of momentum being side derivative with respect to position where tells you is how the wave function changes with angle so images remind you if you look for angular momentum ideas states them with looking for his functions which satisfying minus RIT Deeb ID data as a function of Oregon fade into a images we don't have to worry about 5 hours later they differ looking for an eigenvectors the angular momentum we want to set that equal number they are the Eigen value no time siren fate it's a little differential equations and it doesn't really depend on all our just a spectator here are derivatives with respect so this is an equation about the fate dependents care about orally know what the solution of equations like this is the solution is stiII equals E. I fade Ito the aisles player with hassle at Dell has to be the integer an integer in order that the wave function comes back to itself when you around closed a close curve Well what about the dependents what's are dependent be doesn't tell you what are dependent so the answer is you have an angular momentum Eigen function no matter what what's I use the same function which is a different function squad car off are any lay function looks like Eaton AIL fade it doesn't matter what it are dependences who will be an angular momentum like say so I angular momentum has to do with me angular dependence of the wave function the young the radial dependences be relevant to the angular momentum may be relevant to be energy but not relevant to the angular momentum and the same is true in 3 dimensions of 3 dimensions angular momentum also has to do with the angular dependence and they're a particular wave functions generally arbitrary functions of radius doesn't what they are terms specifically function they usually called Y of fate and 5 which are analogous to the ideal and also wave functions of the things which determine the angular momentum of the state of things which change when you rotate the configuration an angular momentum has to do with rotating configuration so only talk about angular momentum like states we're really talking about these functions
of maiden flight now is awkward to worry about and fight we don't have to we did last time was we used a bunch so go back to those tricks and remind you but keep in mind at every stage what we're talking about is angular dependence away fortunes OK Webster play what we learned last time but we started over the angular momentum operators L V L X Hawaii we found out that they have certain commutation relations L X with L. White Eagles Elzy His age bonds in the New Labour them about right now X Y equals and cyclic permutation like that we saw the play some games we played some games because back for us to play those games not because any of us would be smart enough to figure out what sells Ahmad we would have hours after but the trick was to construct but could take 1 of the operators and work in the basis of that operate another words and work with eigenvectors of LCD with Eigen value and we don't know and what the spectrum of eigenvalues is but let's just postulate watch justifying the Eigen values to be little in forget for moment whether integers or not may may not be integers but this the the eigenvectors of cozy with I get value and worked in that basis and we found 2 are great the operators L X plus or minus I LYT which recalled plus or minus we had some nice properties were useful Pa I'm not going right and the properties you have written down from last time I hope follow properties follows commutation properties that follow from the connotation properties of the elves but here's what we learned we found out that if we start with an eigenvectors villas eigenvalue and and we hit it with L plus years for we would put the pluses and minuses upstairs and downstairs does matter whether that that gave us another eigenvectors we close 1 rumor that that was Carter was a consequence of the simple algebra of these commutation relations we worked out by just show sticking India sticking equations was a couple of lines we found out that l Plus is erasing operated at school erasing operator this is this line of represents the spectrum of possible values of him but you 0 so forth this tells us that we found on another eigenvectors which is 1 unit up there has not yet told us anything about whether and are integers on marked tells us that if we find 1 we can find another 1 by hitting it would help plus likewise we can find another 1 by hitting it without my L classes and l minuses are operators which take you up and down the spectrum of of eigenvalue use of OK that was the 1st thing that we discovered from the algebra of these things the next thing we're will is go on forever and ever and ever it will go on forever and ever and ever unless it comes to an how could come to an end a lake come to an end is if l plus instead of giving another 9 0 eigenvectors gives a 0 only way can come to an end but it does come to an end and typically do come to an end if does come to an end that CA of spectrum raising but to get by raising the suppose at here maybe we however things separated by an integer but we have 1 more thing that we know we know that z-axis that is nothing preferential about the plus z-axis if we simply rotated the whole picture rail and replace the classy taxes by the mind z-axis all the rules will be the same so for that reason the spectrum has to be symmetric about the origin anything you could say about Elzy you could say exactly the same thing about my Zilzie because it is a agree related by rotation so the spectrum and at some point here it has in going downward at exactly the same point except reflected now each action of El Paso might this places you buy integer how is it possible to come to an 18th with the aim is being symmetric like this if they have to be integers for example of this was three-quarters here and you can't going down down down
down down you would never compromise three-quarters right now will you never come true worth yes you never compromise callers killer that can happen with you can a and B symmetric is if you started the integer that's 1 possibility of saying is
integer Julia and then ended minus 2 1 possibility the other possibility is that you start and a half integer for example the simplest case would be cut the spectrum and a half and in the other direction of minors that's still separated by an integer In that case ever plus would take you from the lowest state for the opera state minus would take you from the opera state lawless state but will happen if no plus the upper state do 0 hits the lower state it would give 0 so that's the way we know what a spectrum of the angular moment that the spectrum as you integers a half integers and each collection of states like this if use it is at the highest point which is usually called let's forget they have Richard trees have the educational combat tours of not
are probably not tonight let's take the integer case her and only yes you 0 1 2 3 1 2 3 and only here so the Bahamas states 1 2 3 4 or 5 6 7 states he said possible states the highest state has the maximum value of In another before it's killed by next actually operate at the maximum value and is called El no equals the maximum value of and for that particular collection of states might as well as of course the minimum value I love him and other values a goal from minors L the plaster and they nothing beyond that from above nothing beyond that from the law firm states is out altogether the bureau from my inner self plus 0 2 0 plus 1 so a motor Brett a collection of state characterized rich with charm will be given value of Siew Obama given value of the collective states of this type a multiplex has typically Joe L. plus 1 states L. plus 1 states as 1st thing refined could have see other lack of services illegals affinity Mike right it is a possibility but go sour mood it's not 1 that will a about the pathological Reza a pathological for a variety of reasons the number is 1 of the reasons that pathological but among other things energy usually depends on the angular momentum as you increasing angular momentum keeping other things fixed energy goes up and armored car if you're interested in states of finite energy when you not interested in states with your momentum OK let's go 1 more little exercise what characterizes these multiplex but value characterizes these multiple it's it's the magnitude of the angular momentum all on this states in this 12 plus 1 collection have the same value of the magnitude of the angular momentum no Amin by the magnitude of the angular momentum would mean by the magnitude of the angular momentum is exactly what you would mean classical physics for Russia say the square of the magnitude of the angular momentum it's the magnitude of the angular momentum vector squared and that's equal to L X squared plus Y squared ourselves scored the only difference between classical physics and quantum physics is these now operators the Natchez numbers in the usual sense the quantum mechanical operators let's see what we can find out about ill square in particular L squared from multiple it To well plus 1 state supposedly have multiple which terminates from above that terminates from below what might what can we say about L squared so Is the tricks and Ultrix but nice tricks this is Eagle to Elsie squared now 1st classical LX square Y squared plus L X plus I Why do I want that more elects minus I wiII and LX plus IOY L X squared plus why square and the crust 0 classic so this is a correct equation classically that's not quite right quantum mechanically and the reason is that elects to know why don't Camille we actually have an extra term here we have the extra terms to all I L X Hawaii was prominent money I know why acts so it was the kind LX Hawaii Jerry become a trader
on LOX and L yet With another overshot by writing in LX minus like exports we have to subtract Miners I can't relax a line that's just the extra piece it's there because LX Times is not equal to a white and pink such straightforward but was the commutator of LX Hawaii the Hawaii is ideals
song this is equal to Elzy square now we're going to get an idea from the IOC it is not here and then there's another line from the commutator when eyes plus 1 service would plus els e els e squared brussels and then we have plus L minus Times l plus OK me that's ill squared the squarely angular momentum now lets up applying that let's take this operators enough apply it to them maximum highest state M equals a and equals Del so let's take this L square on bail L is the highest state where we get we get Elzy squared on plus els e climbed l plus a hopeful miners plus Fahringer OK 1st of all what is this script he eigenvectors remember what it stands for its stands for the eigenvectors of Elzy with the highest possible value in this multiplex what is Elzy Elzy is equal little else stayed up here Elzy is equal to little else to score back to me letters and sorry somebody letters but that's just the way it is els e of state am was defying the BN eigenvectors bills set in particular that Top eigenvectors here we just substitute for and we substitute failed so hilly L Z on a L is just the medical number out on him what about Jose square this is you just would be operated Elzy twice the 1st time you hear it gives you a mail a 2nd time you hit it gives you another L so that square but that was easy but what about this year Ferrari former yet the law is 0 White because l Plus With hits the top runs out state I was a definition of a copier the definition of a cop state visit plus 2 0 so we can just erase this and now we learned something that this state is an ideal sector of the square of the angular momentum which I swear destroyed L. squared plus find l or better yet no plans L. plus 1 just to remind you came from a mail squared came from here and came from now squared for sale is l times 0 plus 1 so I wish fine we find out the states in this multiplex here the cops state Papa a highest state there hasn't I get the value of El squared which is L Times l plus 1 the square angular momentum for that state is not little elsewhere almost just Louella square is a little quantum correction here classical mechanics we would expect that the square of the angular momentum would just be the glare of the year of the angular momentum about the z-axis for the upstate instead the highest state is like a state rotating about the z-axis where all of the angular momentum was in the vertical direction that's why has the highest value in the other direction would be smaller OK so we final surprise it's not just all square at times a hopeless war but there's another surprise and this is something this is a nice little exercise is not very easy to organize itself every 1 of these states In this same multiplex have exactly the same value little squares they all have the same magnitude 4 square the angular momentum basically what this multiple it is Is it the quantum mechanical version of a state with a given angular momentum except in any possible direction they all have the same L. squared the same square of the angular momentum but they have different projections onto the z-axis what's no quantum mechanics the projection onto busy z-axis is quantized comms integer multiples OK so mats that's all we know about angular momentum young you see what I understand Edwards said the last fall OK guys now there's a exercise Boca laps good what shown is that elsewhere on the maximum eigenvectors is equal to L. Times up plus 1 another words bank this eigenvectors here happens also to be an eigenvectors of the square of the angular momentum next that I would like to show that L minus 1 also equals L Times l plus 1 Tangelo minus 1 another words that the next state Dale was exactly the same value fails square OK Have you do that well say start without squared and now this with L my next Monday minus line L is a state 0 minus 1 don't want approved but that's equal to plus 1 times tell my on L that's the thing you want approved give anybody gets out approve it the algebra here suggest anything of value or just me but suggests me that we figure out what the commutator of El squared with no minuses imagine we discovered madam we discover a marvelous 5 which I'm not telling you withdraw its troops that l squared minus commute fun we discovered that then I could say that this was also equal trying to prove this relationship with say that this is Eagle fellow minus Times l school squalor right what elsewhere the 1st tour Delia arm if this commutes with us we don't push this through here that quarter times 0 plus 1 will work from this relationship we would approve this relationship we proved that this is equal tell my 5 jail Times l plus 1 which is exactly what this says so if we can prove that l squared commutes with El miners then we can prove that the state l plus 1 is
also an eigenvectors square What about 4 of the next down same thing which just banter we ran out of State Farm he is OK here's the exercise the exercise is to prove that l squares commutes with any 1 of the components of angular momentum Coryell supplies LX LYO 0 commute with any of the components that I will certainly commute with no plus no minus a plus and minus adjust LX plus I of customer Messiah why so there's an exercise proved that l squared commutes with any 1 of these sales arm is always found from the things that I showed you plus the little exercise it to convey yourself is it when you find the multiple at like there's going from below the mind is such that is normal or above
it and no more below it you found not only eigenvectors Of Elzy eigenvectors which Elzy is equal to begin but you also found eigenvectors of the squalor of the angular momentum with eigenvalue UL Times plus 1 way is just the maximum value and Saleh for each 1 of these Little ales there are 2 well plus 1 states and they all have the same squared angular momentum roughly speaking they have the same square angular momentum because they just took the the tilt away from the vertical axis a rough state that's mechanically it's a little tricky but that's a classical analog that uh by rotating in changing newsy component of the angular momentum you haven't changed square OK so that's all we know all about angular momentum would not be raised over them you factorization you that Alex where once you could just stared up muscle and a sign here now what it would be given you exactly the same thing that you had the Ellen FUND minus Mr. right you want to hit on the lowest state instead of a higher state of assigned this would change little workout all work the same way the question but OK so what we found is that there are collections of functions of the angles now these eigenvectors of functions functions of wheat angles why because rotation only affects the angular dependence of functions and what what characterizes them a value will of AT&T not the eigenvectors nosy so whatever they are their functions which have removed subscript and my aunt Izzy component but they also have their also eigenvectors of El squared or in other words they can be characterized by a little gala and they're called YEL pan and their functions are they then 5 Yorba seems a why islands of Fagan fine spherical harmonics functions army units years functions of firing in the city logs on this year of each the plus or minus I know I M I L L face their three-dimensional analogs two-dimensional functions strictly speaking is a function of a circle of the one-dimensional fortunes but think of them as functions for the two-dimensional problem because of the corresponding functions for the three-dimensional problem good can learn much about exactly what they are we gonna finessed exactly what they're whenever they really need we could figure out what these functions are but we never need all we ever have to really know is that for every little value of El Dorado will plus 1 of them the 2 well plus 1 of them have magnetic quantum numbers in that range from 0 2 l and and that the square of the angular momentum is still plus 1 that although ever gonna have to really know the door atomic physics took a questions about these structure of angular momentum multiplex and they all have the same energy they all have the same energy and the reason is because he angular momentum commutes with her more premiere that means when you when your plight L tour of function it doesn't change the energy so they all have the same energy at least for problem of essential for the central force the angular momentum commuter and that's enough to prove that all of these have the same energy OK no questions on the wire in fact that move now to the central force problem now I'm going and not going to take you from ruled all of them manipulations involving partial derivatives all over the place with respect the data and 5 so forth with dough this by using a little bit of intuition from classical physics and work out what you we felt that kind of cheating the cheating is going to be use classical physics to make a guest but the guesses are well motivated and we can know we can do this rigorously was a lot of the we last saw was running a equations for problem with a Hamiltonian which is peace Quella and now he's stands for the 3 components of Mayor of a momentum vector P squared over and kinetic energy plus some potential energy which only depends on the radius central force problems later will be specific about what central force problem of course or think about call forest fire but for the moment but not a snob specialized West or classical physics 1st OK 1st statement the angular momentum was conserve now you could check it you could work it out by working out the equations of motion and seeing the angular momentum is conserved you could also just say angular momentum is conserved whenever you have rotational symmetry and that's correct to say statement as saying that you have momentum conservation whenever you have translation invariant nurses classically quite Pfeffer had ass a share are trip put the goods gone I now but I started a total of Louisville angular momentum is conserved triangle that the accused he not gross 3 its could served that means magnet donors conserved Borel forward store actions Contreras and that means as Aziz said before the of the is also considered don't want to another place the orbit is also concerned that everything is rotationally symmetric we
might as well be classically but this whole thing in any White Plains as orbits do not have to be circles most obvious enough circles in fact for a general problem most orbit or even be close to take a general potential most or matters will not be closed they will process about around the axis of the order will process but I don't worry too much about what the orbits closed now are concerned right now now next let's look at In orbit his R and let's put the momentum he at the momentum as along some direction it's in the plane because we've because we know that the motion is a place of the plane and its pointing in some direction it may or may not be perpendicular Torreon factor could even be the velocity could be a long guard their orbits for example would just go back and forth through the origin and so there's no rule that says that the momentum is perpendicular there is no rule that says the angular momentum is perpendicular to both our and momentum but that's what finished with that OK let's take the media a momentum vector and decompose it into 0 8 angular part but scored the faded that component P data and 80 are component early momentum has 2 components 1 along the angular direction and 1 we are direction What about these square be squared is the sum of the squares of both of them are still H. PR squared plus B status glittered although 2 fanatic or about the angular momentum the magnitude of the angular momentum the magnitude of the angular momentum is Pete at times are clear to you that Pete Seda times are is the angular momentum are crossed Peter Barker recipe the magnitude of it is the magnitude of our times the magnitude of peak b magnitude of the angular momentum is equal to and ah everybody magnet Pete Seda conjure art for another later right it is ills of else squared is equal to squared times or squared now what's right Pete data squared is elsewhere overall are squared was easy let's stick it into the Hamiltonian PR squared must be there was of sorry you didn't correctly plus VOR OK PR square but just PR squared H PR squared off with 2 and what about these say square feet squares GL square divided by square plus VOR what's erase all of this narrow but they have we have a problem which for all practical purposes could be thought of as a problem in 1 dimension namely the radial direction PR squared plus a potential energy VOR Plata and not determine the potential energy remember that Els conserve so want to pick em want to know what the angular momentum is it doesn't change our new so want to know what it is you could freeze it and you consider Hamiltonian of the system is just PR squares so it was just a one-dimensional problem plus an extra turned which goes like 1 of our squared plus the original view are what is system called physically What is it is it is it attractive hours a repulsive at each Delaware attract repulsive what it is attractive potential energy 1 of us could get bigger as you move late so I swear figures you will always be 1 of our squared decreases as you move away so which way is force pushing you forces push you toward smaller energy forces always push to put smaller energy that's why they might miss the derivative of the potential with respect position this is pushing you always from the origin but cast things is pushing you away from the origin there was nothing in this problem the pushed away from the origin were that introuvable force or This is the potential in a that's associated with this introuvable for and it gives very big targets when I get small fixed angular momentum of fixed angular momentum really repulse you away from the origin and so far Humpty comes in with a given amount of angular momentum 18th get very close to the origin but does a swing around the origin OK so this issue but keeps the particle from getting to the origin is this the local forest is only 1 situation where it's not there when is it not relic was 0 the relic was 0 and then Fraley equals 0 we're talking about or which just goes through the origins back and forth and back and forth the origin has no angular momentum and this term wasn't there for as long as any angular momentum and it can't get to the the origin because as huge
repulsion but that's not a real repulsion that's the centripetal force birds their peeled what about of artists system momentum associated with the art direction the school and the 1 who will more will be more with classical physics let's plot this effective potential energy this becomes the effect of potential energy for the 1 individual problem it has the ordinary potential energy sources tropical barrier this is called for Vogel barrier let's supposedly of ours attractive for example it could be the attraction of electron proton power then Warabrook something like that still a little better VOR it would be negative because it attracts and it would be 1 of are so it would look something like there's no other that at what about L. square along square depending on whether or not to any L except illegal 0 it's going to blow up at the origin like 1 of are squares and then fall quickly 0 the fact is view of is or potentially goes like 1 or are it falls more slowly than the introuvable barrier introuvable barrier falls very quickly 0 1 of our squared and wall a little more slowly was a combination of them look like well a combination of them which is dominated small are a small our which was big 1 of our squared off Laura square much bigger sold power very close to the origin this is the dominant thing What about a large or if these 1 supposing musical long potential 1 of ours bigger than 1 of our squared sold emphatically far out whom potential winds what happens in between what happens in between looks like theirs OK so what kind of emotions deal we have an effective problem which is completely one-dimensional now always depends on they are accidents and all of the angular story is fixed by the angular momentum want to know the angular momentum it becomes a wonderment what if think motion in this kind of potential looks like well the simplest kind of orbit was simple as car motion would just be equilibrium at the bottom of a potential that's it orbit in which are doesn't change its a circular orbit OK at a circular orbit where the kind of orbits can you have well what emotions could you have it a potential which looks like there's basically you can't have oscillations back in Fort oscillations which go back and forth take the particles from a minimum distance from the origin to a maximum band back again so these orbits which go from a minimum distance to a maximum distance and back again of equal potentially would be the elliptical orbits and that's the classical physics in a nutshell classical physics where about the quantum physics a quantum physics we have a wave function wave function instead of don't have the particle position we don't have momentum we have wave the wave function it is sigh of armed and also angles but the angular structure is public show account by knowing the angular momentum if we know only angular momentum then we know that we angular part of the wave function is just 1 of these circle harmonics which I erased but we are afterward it's a one-dimensional problem sigh of art and wasn't satisfied satisfies a Schroedinger equation each on side of our let's say the time independent Schroedinger equation was a time-dependent Schroedinger equation 40 per right inside the energy that it determines the Eigen values and the eigenvectors of energy no 1 is 8 begins with PR squared was PR PR is of course my age Bar B by as always quantum mechanics a component of the momentum is each bought kinds derivative with respect to the card a corresponding court some PR squared his PR squared PR squared is going to equal minus because of 2 factors of MORE 8 4 square and bring in each bars narrow 2nd derivative with respect the ah that's the 1st the Hamiltonian and of something else I left them twice the maps it was best not be talking about incidentally the planet Mercury will likely be electron for this discussion but it could be the planet Mercury to head out of the with Bob doing quantum mechanics early a 2nd by school and that's a 1st turned Hamilton the 2nd term in the Hamiltonian is exactly what's written there plus 1 or 2 and L elsewhere are squared but what is El elsewhere that we come back if L squared is not a little else squared ill-timed plus 1 white because this is the square of the angular momentum what appeared there was a square with Ehrlich square why square squared and so has becomes L gun jail plus warrants Omar squared that's 1 of the effects of quantum mechanics just to change those squared tons of plus 1 and finally plus of
ah you comply Binya favorite potential into their where we do it week at Durham side sigh is now a function of aura side are sign of sigh of armed equals he saw throw your equation solutions about equation are the eigenvectors of energy was given angular momentum and some of the Eigen values are the energy levels I have a feeling I left out some each Baer squares and he alone probably did because where did it come from the king from he probably age Glidden here to track I think yes but that's like to solve would like us solve this find the energy levels of the move magnifying the BEL wave functions of an atom would like to know what the electron probability distributions light and so forth OK so what I could do that when I could do it because it's just a matter of solving differential equations and this is not a course in differential equation but we can still get about yeah all you all of them yet all of all of those all of these levels here the Tulip plus 1 of them have the same value of Lu L and synergy your readers its where the dollar but it side we're combination of and I was 1st to their a just not only angular momentum and we know the angular momentum quantum number of the wave function that we comply but it's tonight to remind value for that multiplex there are wave functions of any angular momentum but they come in knees in the pattern is that they come in 2 little plus 1 multiplex tour plus 1 Moapa blips on quantum mechanical analog viewfinder wave function and you rotated around this tells you what the energy levels associated with a given LOR and they want a sovereign al-Qaeda delight to solve the most potentials most potentials right guaranteed have to solve the medically the Coolong potential happens to be solvable if you want solid solutions of this equation are called Deegan Bauer polynomials begin malfunction good luck worker bees from you who were very little from that I use the gig involved about Holland was a demanding OK let's look at the the effective potential we're doing is was solving the Schroedinger equation in a potential whittled side bets for solving the Schroedinger equation a problem which effectively has a potential energy which looks like how we want to know what kind of wave functions on what kind energy levels would be OK if you know a thing about Saab ensuring equations then you know that even a L Jewish given was given the potential you know that 1st of all typically there are more than 1 state abound status state which is complying with the wave function which doesn't leak out too far they correspond to discrete energy levels and you know that's the thing which characterizes the energy level is the number of nodes in the wave functions as a ribbon nullity is a wave function which passed dozens which is never 0 what's it always positive has normal if it goes up got them passes to the origins and has 1 so here are some functions with different numbers of no here it is a function with no here is a function with 1 knows about a function with erodes folks in with 2 loads up Coombs goes backdrop from the number of nodes Our what characterizes the different energy levels a solution to the shortage equation the more nodes behind the energy why would be that the more nodes the high energy what yeah knew something like that the monopoles Knowles the faster we wheels the fester illegals dot the larger the momentum momentum is characterized by Whittle's the wave functions of momentum states will fire the momentum the fair so they will go the more they will the more nodes that have so sold generally speaking it's a furor exactly the lowest energy state in a given potential has known the 1st excited state has 1 no next excited state has to knowledge eventually you may or may not run out of a of our solutions but a mail order according to the number of nodes so let's start at sea now we can say about the spectrum Adam it doesn't have to be realistic Adam this could be an ad with the aware of our potential like to gather it could be 1 of square of art could be eaten the miners are some potential what we say let's see if we can organize the energy
levels and the states of the art
OK what's your removes the Olaf plot horizontally the different angular moment With the Willie L. real AL was Iraq 0 Kelly calls 1 too illegals free so that each 1 of these els has owns equation remember the Schroedinger equation involves bill as a parameter so each may have several bound states several solutions let's think illegal 0 helical 0 will have a lawless synergy state Matsuo applying Hewitt's plucked Energy Energy vertically al-Qaeda be the lowest energy level it has no knowledge of helical 0 so someplace over here what's the next state with 0 or the next day with illegal 0 has a node in it said potential 1 no but up here somewhere summary states are there for every 4 helical 0 each with each case has won state health plans plus 1 is 1 when I was equal to 0 ranked 12 plus what it a total of 2 Toccoa was 1 is ready to help us what was 1 so every 1 of the states with equals 0 1 nose to knowledge Reno wasn't mode they all have 1 state associated with them and some but not equally spaced there or whatever they are whatever they are Rosalie energy levels With zeroing in momentum now looks go angular momentum 1 I feel momentum 1 there is also a state with 1 with no not so we solve the same Schroedinger equation except with equals 1 year at times outpost 1 when I was a good 1 1 times 2 is to right went on to lose to the other hand Felicle 0 this was Iraq saying Schroeder equation different value of and illegals 1 that will also find things with 1 knows with their energy be larger was smaller than a L equals 0 state with no no just as wild guess but angular momentum midst of swinging around their it's got angular momentum and besides which this is a positive turn in the Hamiltonian possibly lower the energy so Kelly pools 1 state with no Woods will like it someone above me at the corresponding state with no equal 0 likewise became a state with 2 nodes will lie above this 1 and just
by virtue the whole spectrum may move over but it will move up a little bit because the angular momentum was higher How many states are buried each 1 of these levels drawn a real Dashiell they indicate a possible energy level 11 a doable but was it be generously Hanley states are there each 1 of those dashes to 0 plus 1 3 so all we should really drug users 3 little bashes if I was trying to separate and indicate 3 states he of these is a trip like What About Elly equals to and illegals to we put in a little more angular momentum truly we've raise the energy again and that we will have somebody states would be for each state to 5 2 0 plus 1 song Delic was to its more more little bit illegals to we will have more up a little bit more have 5 states 1 2 3 4 5 . 2 4 5 and silicosis it was 3 2 plus 1 virtually frequent was 7 states and so forth and that's about what it looks like that's about what the general solution looks like 4 0 so arbitrary potential at some point when the energy gets to Heidi just like in classical physics what happens when the energy 2 for an orbit around the sun an escape again exceed the scale escape velocity of along abound we a restart likewise when the energy gets too high electron escapes and serve about the energy of appear no states above their actually back Saul abound states have basically negative energy energy stocks is should be those negative down here and this is what a spectrum of a of anything in the central force looks like Be Coolong potential has a very special features it's almost accidental it's not exact it is an exact statement about the nonrelativistic along potential but is not an exact statement about Adams they where tell you it's violated by every possible correction of the year of the simplest theory of the Adam diets violated by the finite size of the nucleus violated by relativistic corrections is violated by the fact that the electron spin which couples to the everything and on earth violates it so it's something of an accident is very special to long forests and it goes as follows the 1st level here which is moved up a little bit 0 nodes state 0 notes state fail equals 1 happens to be have exactly the same energy as the only 1 node state for L equals 0 he is a one-note state Felicle 0 Zero mode state the first one that occurs Felicle's 1 occurs in exactly the same energy as the 1 node state equals 0 that's a mouthful what about the tone or state the 2 notes sorry 1 state of equals 1 occurs at the same energy and 2 Norwood state of Ehrlich was 0 exactly the same exactly the same for simple Cologne and so on and so forth this is apparent that you go to LA caused to be equals 2 0 0 knowledge state in other words the lowest still equals to state has exactly the same energy has which is novices this is not our war toner awards appeal of 5 states 1 2 3 4 or 5 about BLE causerie cost 3 begins PS 7 states from 2 3 4 5 6 7 and soulful that's a pattern and it's an accident it's not completely in accident nothing in mathematics is ever of taxi but you can't say things are accidental when you're just small changes in the potential for small changes in anything near a ruined Anderson absences annexed I'm not going to work I'm not going to prove it it sounds like an extra symmetry Southwark extra generous it sounds like they're generous he's between states is a symmetry but rather bizarre 1 it's a rather bizarre warned that you would never notice if you were thinking about this very special political on potential and is not exact so I think we will pass proving it it's a nuisance to prove armed but it is a fact very special fact about the special war while our potential so there that's the spectrum What is it be generous city amid the generous immediate generously at each level he is the lowest energy level is only 1 state hit has angular momentum 0 it's called on grounds that was the lowest energy state of the art if this were the hydrogen atom for example will be the most deeply bound state that you go up 1 level you go to the 1st load of the Zero angular momentum state or are known notes for the angular momentum humbly states did get 1 this is a form of the network 1 2 0 3 4 or 5 6 7 8 9 9 states to the next level and we're on a guest of the next 1 is 16 is right so for the cool on potential for the cool potential the states of generate like this and the D generously is always the square of some integer Adam a proxy hydrogen and approximately satisfies Fatah a good approximation but all kinds of things violated relativistic corrections and as I said just the finite size of the nucleus of the despair now the nucleus is very very small wine as a finite size of the nucleus disturbed at the fire the nucleus is a little ball charge we pretended to charge was a point at the center that we give us exactly the call potential of the charge of distributor over a little more potential energy would be slightly different than the exact especially when you get into the nucleus electron can't pass the nucleus of just the ball charged and the Place adjustable charged electrons held sorry it attracted to the nucleus and can't the the nucleus the passage of the nucleus its Sanchez de facto of the charge distribution is not the perfect 1 of our square at 1 of our potential the potentials anything but won't OR misty generous he fails sober costs the nucleus has a fine I use it his vary very slight deviation away from us now the nucleus is a 100 thousand times smaller than the size of the atomic wave functions of the nucleus is very small and this is a very
negligible effect but Gillett point is just this is something of an accident and not particularly deep I don't know anything about it I love the fact that has this kind of be generously and that spectrum of the hydrogen atoms that's a spectrum of a hydrogen atom in idealized approximation nonrelativistic perfect Coolong potential 1 state the ground-state for states above it 9 states above that is the way Rio Adams work notes about 40 real Adams were real Adams To generate cram states 8 1st excited states 18 next excited states 32 next excited states sell something missing in this analysis by a factor of 2 sped back so we haven't talked about spin and I think we who won't tonight that's all of this is of course just factory for solving the Schroedinger equation that may write the Schroedinger equation in its original form before we went angular variables it would be particularly helpful to try a solid but we might just try work the writer M P squared means PX square PY square was busy squared it could also mean PR square 1st square is PXY opposed BY so this really means minus once bar squares these 2nd with respect X squared plus be 2nd with respect Why squared 2nd with busy square off sigh plus V R time equals Eastside equation we really set out to solve solve it by going 1st so Pola Court middle radial coordinates radio angular coordinates and then using algebra Of the angular momentum could deal with the angular parts of it that is very simple the angular parts of it are all subsumed by saying you work with angular momentum Icahn's states L. squares finger comes in and squatters nothing but l times of plus 1 by that time you're dental one-dimensional Schroedinger equation and you use whatever mathematical work strategy you know for solving such things I want to get bent on 1 direction it's just an ordinary differential equation it's not even a partial differential equation you only have the derivatives are so their worst it's a numerical problems us give me a lot more weight size nor nor number it too many grabbed a correctional too much bigger not I just point out that beyond the nucleus size because it's an easy 1 to think about but it's it's minuscule not a lot of can go elsewhere is visible and about this 1 now was that would permit capital he also square Times l plus 1 young men the less you know that's in other words it's a little quantum corrections a quantum correction young no not spread but they but they fall off very rapidly as you away from Compaq support would literally mean they go to 0 outside a certain distance statement that the wave function as can't support would mean 0 out as point but it's exponentially small think he'd be decreased exponentially sold for practical purposes if you go out or surpassed a certain distance each wave function becomes negligible small into question arbitrarily far-right but it goes is your exponentially with distance no 1 else 0 nearly half of its value and are good that's actually that that's right when l is 0 then introuvable Barry right the state we point the origin of young backtracked classically it would just go to the center or just go to the center and the center quantum mechanically it can both sit at the center and I would go to the center not only goal of the center but it would be be arrested the sale lowest energy would be at the center where we have to get negative in the incident energy you would lower the potential energy by getting to the center and you would lower the potential energy by making momentum 0 you can't have balls you can't have the the momentum and the position definite as a compromise a compromise is from a wave function power with a little bit of spread away from the origin here we it should be a T-shirt anger equation has a solution a solution has exponential exponential pale but concentrated near uncertainty principle which prevents it from sitting literally sent him daily for let no what no not OK this is motivated by classical mechanics of 2 of classical mechanics motivated Peace square Alberto Lim more P was a vector and allow us to write this down once we write this down we have a very very precise quantum mechanical problems exact Schering bombs I did a little bit of guesswork in going from year to year classically it's completely rigorous to go from PX square post PY square was busy square good PR square feet square you have to justify overstepped quantum mechanically but that's that's something Whitacre an hour at the blackboard you would not you have a seat but by those classically classically consequent quantum mechanically the wave function fills up everywhere from pools of 3 classically and the trick is to say angular momentum is conserved that means the plane of the orbit is considered just change coordinates so you're XY planes in the plane of
the of the orbit Gato that are mechanically it but I don't know but the fact that you can do it and believe it or not just a statement that ill squares replaced by plus 1 critic but it's it's these other LX is analyzed creeping into a gap if you could just put in a fur squared Elzy square that it would just be a little elsewhere the fact that its three-dimensional comes back at you just from Yale and of course Laura all right not only will you very lucky and I would say basically physics of lucky this way twice but however the harmonic oscillator piece of luck is very pervasive almost everything in nature to be well approximately although many things in nature and be approximated by harmonic oscillators for example he then a young not illegal Zero would go to a higher angular momentum states in the hiring a woman from states a potential looks like this and they're all orbits which oscillate back and forth the bottom here and and the simplest approximation of those auditors to treat them as a harmonic oscillator all kinds of places were Vermont oscillators because of that the trick of whatever trick whatever trickery goes into a harmonic oscillators very pervasive but that there are not many mathematical cases where raising and lowering operators or sometimes a call ladder operators only when you're very lucky OK all that that doesn't offer Adams for the moment with the comeback that Adams Spain would talk about the fact that electrons affair beyond that both on and how that determines atomic structure of bit power and Al Adams radiate and give all photons quality stuff will come back to it but let's forget Adams of time being come to the harmonic oscillator but I said everything in physics which has an equilibrium you disturb the equilibrium a little bit chances are that behave like a harmonic oscillator so it's a very pervasive thing and model for it adjustable the particle or await them in the spring farmers but a simplified a little bit of some of the parameters aren't terribly important will a magic awake hanging by a sprained it has an equilibrium position and the deviation from the equilibrium position not deviation from a from where hanging from the deviation from the equilibrium position will call acts of God take the massive To be worried that wicked we'd do it for other messages that doesn't go a little exercise to delay of us very is nothing new nothing new happen so the vessels 1 and the spring constant across this spring constant Katie Katie is a standard terminology for spring constant will call Omega square forever find out the moment the may gives the frequency of oscillation of the spring constant as Omega square where the energy of a classical oscillator began Aegilops call give its name the home Monday Hamiltonian Hamiltonian it the usual Peace Corps now he is one-dimensional just a momentum the X momentum p squared off to an P square or and as the article plus Katie Omega square X squirrel 2 what's worked out was just checked this is right to do was worked out Back at the equations of motion if you remember Hamilton's equations of motion so pale opens Hamilton's equations of motion D H by a D D P equals X . each by D X equals miners P . 70 people remember my P . thank you to remember that because you go the village jackets you we get pH by DP is equal X . pH by DP is being so this says that people's sexpot right Fesub Tibet momentum as a total velocity but at the Mets awards at the mystical toward that's OK so I am a physicist among the musical backstop that's good now what about BH by DX DH by DX Is Omega square X Gaza martyrs Vargas OK let's check Newton's equations as calculate the acceleration the X bubble bath as P not P . is equal mark P . renewed pH by the X Nichols Marines P . equals plus Omega x squared each by Exs plus Omega x X double . is equal the commodious Omega squared X X double bottoms might Omega square next that's equation of an oscillator with frequency omega theory recognize text of about minus Omega square backs acceleration them was equal to 1 acceleration is equal to restoring force for it as I said cazique Opel Omega square OK so this is the right Hamiltonian furry harmonic oscillator and now will those find all its energy levels eigenvectors and find out what it does is quite the mechanic classical Emina workers swings back and forth with frequency omega OK let's see what we can do do it it's a sum of squares 40 below the summer squares if factory everybody narrows effective the summer squares well not everybody knows that Iraq might so let's write 1st of all one-half downstairs and that's right P Plus I only get X times he might are Omega x loading familiar not from the harmonic oscillator but from about an hour ago LX plus IOY Alex minus wiry sorrow let's do this business right now bomb also about a young women lawmaker downstairs and
Omega upstairs all week because its convention OK now Billy not propels a makers or that later His a tricorn mechanically not because as a true key times X and X Times p but they don't canceled because Exon P don't commute so let's see so I have to subtract here after subtract minus 1 half from he and and then I will have alright now we have Omega X Times P minus peak times acts 6 times P E minus Pete plans X and I think I think I have a right subtracted it because it was appearing in here and Lynn on wanted weight so subtracted it and now I use 1 more thing this is the commutator of accident he wants accommodative Wrexham pink I have dropped beach bars all over the place will put back beach bars later OK XP minds PX that I can't dictator PT of X was pity angles are strictly IEH each borrow from wanted to keep the yield Scott said while you remember from 2 years ago that's the basic canonical commutation relations of the court and the corresponding movement in we use the new Dow one-dimensional case no matter what 1 In Mrs. this this is Exon PEX is only one-quarter of the problem here are a dimension deployed don't look at is a one-dimensional Monica oscillators sewers XP miners PAX let's keep each around for a 2nd hour forum for 2nd with Iran will drop by each Miners I hide plus so this becomes plus aged bar may go the the OK this is a constant This is a constant what a call Skyrme 0 . inventory it's very even now when the harmonic oscillator is ground state see that even in the ground state where this is 0 everything else 0 this still remain why is there a ground state energy bill but that's as white as when Mark Astley battleground state energy is the energy 0 what's the energy of the gram classically 0 has it get to be 0 well the ground-state is just a year oscillator at rest people 0 apposition 0 both Luisa 0 quantum mechanically you can't have both X and Pb 0 simultaneously furthermore both of these a positive and therefore the king cancel each other so there's got to be a little bit of energy in the ground state if the Heisenberg uncertainty principle which tells you that you couldn't have had 0 energy in the ground state for cable want you know that here let's the Hamiltonian is this thing over here plus a constant but struck back with the Council may not be important but just a additive number every energy level has and extra H. borrow may go over to when those anything terribly important that we drop on drop it but later on we can come back and Back in the overall air of calm script energy now say a further by the back at that could have been negative because backswept P squares positive the energy is X Clippers be square the positive definite operator command negative but so let's concentrate on this thing over here let's forget where it came from a forgetting where it came from now we just want to know what the properties of this operator here's a certain operate it is a Hamiltonian apart from the center among us they want more as an editor of constant and now say again until I put it back eventually but this is a Hamiltonian Vermont the stillbirth spectrum is words eigenvectors always it properties have tall operators factorize is very nicely makes since to give these operators and would do something just for future benefit I'm going to put in all make up here and I'm put a square root I make His older than oppose square root of tool maker downstairs and the square root of 2 Omega downstairs here we see what I did I took the 2 of those downstairs on but it between these 2 0 at the Omega which are stock upstairs and Split between the 2 factors again this is a standard way that even the classical harmonic oscillators often written in this way we need to get these guys P minus sigh Omega x Over square to Omega and people outside backs their role is going to be exactly the same as the raising and lowering operators India and angular momentum case it's pick us up and down a ladder this time it'll be a ladder of energy levels on a ladder of angular momentum OK so let's give names 8 class is equal to P Plus I Omega x squarish tool maker and a miners is the same thing with a minus sign square will make they are permission conjugates of each other missing conjugated each other as 1 plus I Omega Exmore sire Romania acts I think we can now right that the Hamiltonian is Omega times a plus a minus that it is really simple except for Homles was buried in plus and a minus 0 that's a crappy looking thing sure square roots of Omega square of 2 0 but it turned out not to be so crappy what what do we do with these forces of a normal who would Divac do yes I do that that that it would probably stay out of Prilosec decade figure the I values
Gaelic argued that would commute then you would say what the come later here I have had not found in which is a plus and minus I don't know much about it blossoming minus the let's figure with what the commutator assumed on board that nothing else to do with the electors and like the sitter and commutes pigs OK so let's let's take the commutator Our 18 miners with a plus that's going to be commutatum above now is going to be a tool maker downstairs right downstairs 1 lawmaker and then there's gonna be the commutator or peeling asylum X with I Omega now hope commutes SoHo here doesn't commute with who doesn't New with home cracked Pete doesn't commute with X an ex doesn't commute with that's all X commutes with acts and Pete commutes of so what we have here we have equals 1 over to lawmaker now we have I only what's stake the stake but this when he 1st we have a might Omega Omega From he and then the commutator of X with P E what's the connotative X was Pete body so that gives us a Iike terms minor plus 1 we have an Omega overall may altogether this is just 1 half but that was only 1 turn that was this commentator on this 1 with this 1 the other 1 is the commutator of PT with I'll make it acts it's exactly the the same as you work it out you'll see it's exactly the same number so altogether you get to wear them plus a half and the commutator is warned we now know 2 things turned out this is enough is really all you have to know you throw away everything else Hamiltonian is Omega a plus a minus and the commutator of minus with a plaster is 1 in this would fact here and it's the fact that a plus and minus permeation conjugates of each other that plays a role OK let's give this thing a plus minus by itself on me but collect capital and leaving it simpler now the commutator they are buried His Omega times In the end the next step of course if we had the through a Dirac would be be to start commuting in with a closer in with a minus it's a good thing to do that to the thing to do is find out the commutator algebra is very simple but I don't think we have to do it I think we can now proceed without doing what see what we could learn 1st of all this In the object is permission operator if you take operator plans all her mission conjugated always her mission that's something you could check that we in the has a complete set of eigenvectors off course in his the mission basically apart from the fact May gives the Hamiltonian started mission it stays so in is evolution operator and that means it has a complete set of eigenvalues and eigenvectors we wanted the was fined those eigenvalues and eigenvectors those eigenvalues and eigenvectors of the energy levels and I states of the of the Hamiltonian OK so let's let's see what we can learn we want defined the I values of game once we know the eigenvalues event we know everything we want so what's a policy that and cousin eigenvectors called Little and Namibia Eigen value is just little him I haven't made any assumptions about whether little and an integer or anything else I've called the end because it is an integer but we have to improve that let's just call it will begin is an operator it's a plus a minus little and is an eigenvalue This is the eigenvalue equation locate and we can also write another way we can arrive a plus 18 miners on will end his rule and engine had just written that begins is a plus in these 2 equations overseeing thing now let's take the commutation relations much right about a minus a plus or minus a 9 8 plus a minus equals but let's take this operators equation and operate on a state end and she will we get you remove his equation it's just a commutation relations a minus say plus a plus and minus equaled 1 but then I would say that Ford you're will now and multiplied from looking I'm staring at China figure or to do with it I know what I really wanna do with what I wanna do with it is I want to apply a plus when I want to do is show that 8 plus will increase yeah I wanna show that a plus His erasing operator so what I want I want to concentrate not only in what 8 plus times in so I have to multiply plus this in my goal this show that a plus times into something interesting erasing with a plus operated something however equation with we get on 1st of all we have a a minus 8 plus the 1st turn here on and what about the 2nd minus 8 plus off what's a plus a right service was minus a plus or minus and No 8 plus kind and I written that a plus a minus on in just his numerical factor in he I still have be a plus naturally he finally on the right hand side were have a plus pound OK let's shift this Over to a weapons of my shift is over right here I have a plus times in in here I have in the country plus will the aid plus 1 more where it's good but see this 1 shifted over to the right it gave me this is the only 1 about this 1 here
a plus 10 a minus is the operator capital land this is a state that I am interested in still and on this state aid plus money and circle around that is equal to 0 in plus 1 crimes said state sir I felt pretty much what was expected that the raising operator give me a new eigenvectors of capital and with lag in value when you are so is or found Victoria Way defied plot on the vertical axis here yet I get the values of in then it if I find 1 will enter then there's another war above and more about it and another 1 bothered unless of course I run out but we do prove that we can now is at or above it in another 1 above it each 1 being shifted by 1 unit and an and 1 and plus to itself of I find a solution and I get value in the be another 1 and plus 1 the below-normal plus to storm plus 3 and furthermore you go from 1 to the other buyout operated a plus 8 plus takes you from an eigenvectors value in Tour Megan value and plus 1 so full of these of course have higher and higher energy because the energy is just Omega times in warm-up going down and had a down or a a picture a mind sorts a straightforward Maine the proof of that the same deal that a Sunday in takes you down How followed them take you commit take you indefinitely for that because the harmonic oscillator Hamiltonian as part of its only got positive terms you cannot get negative value the operator it's a positive definite operator which means it only has positive eigenvalue is in fact the a theorem that you multiply and operated by its own conjugated that's on her machine conjugated eigenvalues values of or worse 0 4 0 so it team go indefinitely down there's gotta be last warned that last 1 is called the ground-state is usually labeled all Kinkel indefinitely Baron was at me what taking there was a minus Bond and ketchup and plus 1 and minus 1 or it takes you to 0 for it just kills estates of his nothing below it was the only 2 mathematical possibilities a miners picture down you just run out of state there are most big boy that's what happened in my the final just as 0 so there's nothing down below and equals 0 1st the state corresponds to give equal 0 equals are that was here that must be mined material on 1 those I'm OK sought to Kepplewood I been know will be careless written that a plus means His Eagleton plus 1 but I haven't discussed the normalization of these states the normalization of these states is not end and equals dealt in him was a numerical factor is a numerical factor and if we want to keep track of that numerical factors quite tired now I think we have to quit but there is a square root and plus 1 he times and plus 1 now tried Adrick against and I think it's also true that a minor and I think gives you square root in and minus 1 defined by mistake and I think that's correct Your Europe to your tried you driving a plus Monday miners on it was there was be somebody as OK we get our a game like this on a hand his knee and a plus some square root and in minus 1 now what happens when a class hits in minus 1 year just below the square root in this gives me 2 factors of scholars in so I was a little bit sloppy I didn't tell you what the normalization of the state's were the states and not normalized member me and not her chronic normalized normalized to some weird factors with square in when we need that we will meet it who will write down by a year or so it's getting a little too late to a fall possible details OK young college react to want pouring on notice that this equation here says that when a plane this hits the ground state it does give 0 yuck 0 0 work no it has to be 0 young well the Ovitz see none young have Wickham program over tired now approve prefers preparatory question it is the answer is 0 the answer is the spectrum of capital in here is all of the integers starting at 0 so that tells us that the energy spectrum is Omega plans in integer there is an age foreign here I'll go each Basel poorly with each bombs a hiding next time but down thousand the energy spectrum is Omega kinds in integer plots the famous half each borrower maker which relate to the ground state energy plus on top of it the spectrum of integers based levels that's the harmonic oscillator and found with it food play with more but more important with use of force solve some important from for more
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Titel Advanced Quantum Mechanics | Lecture 3
Serientitel Lecture Collection | Advanced Quantum Mechanics
Teil 3
Anzahl der Teile 7
Autor Susskind, Leonard
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DOI 10.5446/15096
Herausgeber Stanford University
Erscheinungsjahr 2013
Sprache Englisch

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Dauer 1:57:35

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Fachgebiet Physik
Abstract (October 7, 2013) Leonard Susskind derives the energy levels of electrons in an atom using the quantum mechanics of angular momentum, and then moves on to describe the quantum mechanics of the harmonic oscillator.

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