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Advanced Quantum Mechanics | Lecture 2

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were in the midst of talking about symmetry symmetries are operations that you couldn't do wanna system which don't change the description which starred change of which don't Cheney energy levels which don't change the values of the energies of systems that think Jim particularly don't change the values of the energies of the system examples that we talked about for translation if you take em Adam with given energy and you move from 1 place to another the energy doesn't change doesn't matter which comic state he study is not just Her ground stated you French over every of comic state and sell the Hamiltonian doesn't change when you move things symmetry ensuring their equation doesn't change same Schroedinger equation same thing is true for rotating the system with talk a lot tonight about rotating system bombed another example is interchange into particles 2 identical particles you might call 1 electron electron 1 the other electron electron tool icon Harry and Fred Our changing the leaders of the electrons into changing electron 1 of electron Till doesn't change we are properties of Adam that also symmetry we cannot think of it as a cemetery just because was so used to it the identical particles are identical so it doesn't matter what you name them but there's a cemetery and there were other symmetries of physics off for example of the crystal lattice of crystal lattice has a kind of translations symmetry are not translations symmetry and space it is translations symmetry in space but a different kind of translations symmetry of space it's just translating everything by 1 the latter assumed that we have a very very long almost infinite them Crystal minutes almost a perfect symmetry to translate from 1 Adams familial from mom to translate the whole system by 1 letter spacing of course there it's not exactly a symmetry if the system is not infinite 2 systems that infinite them about boundaries the of the crystal well you can't really translated but the crystal went on and on and on forever the you could do things you could a magic taking 1st and moving the entire Crystal a little bit yes as for almost a year the most here the will stay here stay here that is a symmetry of a crystal the stroke the symmetry of anything the ability to move from 1 place to to another but the crystal has a different symmetry which is just a 1st translating from call this 1 2 3 4 1 possible articles I've gone forever and ever just real labeling the Crespo so that this 1 was now minus 1 the swarms called not warm earns 1 0 1 2 3 crystals on run that also symmetry translation of crystals by the crystal lubber crystal lattice spacing Willis what the world symmetries are very robots robots but some of them a more abstract members physically intuitive warms tend have to do with properties of space the homer due mainly of space affected every point is like every other point is translations symmetry the isotropy of space the fact that every direction like every direction is rotational symmetry now another concept will see is related to cemetery not the same thing as Colby generous C D generously CNN specifically be generously of energy levels be general energy levels is simply the statement in energy level and set up a system is said to be generate it more than 1 state with the same energy level than that energy levels called be generous when doesn't happen for a random system with the generic Hamiltonian assuming that a spectrum of it is discrete spectrum some sort of discrete spectrum arm of energy levels it would be an accident rather on access to levels would have exactly the same energy exactly exactly the same energy they may have the same energy the 1 part of the 15th but say they have exactly the same energy what kind of coincidence with that take well as will it doesn't necessarily take a coincidence it takes symmetry in fact those are the only cases where we really believe that they're a true begin to be generous use exactly generous seasonal nature when there are symmetries jealousies sometimes not sometimes not be generously sometimes means symmetry Delaware symmetries some be generous are not always shows them begin with an example in which you could see so intuitively how cemeteries and wind symmetries plight generous that is evidently Bentley is 1 of the meet users of symmetry tantalizing energy spectrum of systems and see that they have a levels said that exactly match that can be very important in absent extremely important let's start with rotations symmetry Buchfelde rotations cemetery and I wanna think about a very simple system namely a particle moving on a circle it could be a particle moving on a circular wire and wanted to get off the circles or particle which is kind of be moving on circle 1 of symmetry the symmetry rotational invariance only described
location on particle wealth if we embedded in two-dimensional space than we could describe it by an ax and white neck among the particle keen get off the circle of its only moving in angular space exclude that it's a little bit excessive to think about an X and Y coordinates we can just think of anything of coordinated arm angle but particle has a way of function used particle here it has a wave function the wave function could be thought of as a function of X and Y E said that excessive Camacho too many variables Wigan just think of it as a sign of the wave function school or 8 times its absolute of it times its complex conjugate the modifying the probability of fine particle on a circle what is the operation of rotation due to a rotation counterclockwise rotation counter-clockwise 6 Starlight side of data minors Epsilon where epsilon a small angle of rotation constantly get confused about whether it plus epsilon reminders that so I hope I have a right but not a worry about it just means almost signs will be wrong if the site goes to sigh of of fate America's epsilon or another way to say the same thing is that change in society at the line of course is our a small number which is so small that square 0 2 the customer number the change society is equal to the derivative of society with respect the data 1st Epsilon I can we were right there at errors Miners i Epsilon kind minors epsilon minors RED derivative upside with respect they not put those I is in there and be call us I want you to remember well let's think about ordinary momentum would momentum is described by an operator which is mine all ID by DX angular momentum it is simply basically the same momentum except week or more admit his angle instead of being a position they operate a mind I by deflator some operate at linear operator are mission operate at the or is there to make it her mission about I would not be her but for resort realize what but fast do I have a minus item I think maybe plus receiver you wrong now I AREA hatchery right my America's I had a right at this time of the subtle from yard of love it's it's correct grammar book it but minors I by these days is in remission operator you may want you may be surprised that hour i in front of for to be in remission after all the mission as the analog among operators a B as being real but I guarantee you that I has to be there go back and you know still of the study of momentum and you find out why that I has to be there let's call the angular momentum there's usually denoted y el don't know why 0 why P from momentum but more P from momentum but there was the angular momentum and so we can write the change in society under the operation of infinitesimal rotation is governed by a minus I can action of angular momentum operator on side that identifies according to definitions of last week that identifies billed as the generator of rotation was if you can't remember the precise definition let's just say that this is the definition of generator of symmetry operation if this symmetry operations is such that has a version of its arbitrarily small parameter in this case angle than the definition of the generator is just in terms of the small change in the wave function when you shift by an infinitesimal amount and that will just give you the a generator don't at a loser Epsilon Hey right I did lose an Epsilon and mystifying L & L is my ID baby OK there was known as the angular momentum now before I go on put in some each bars our our truly dropped them again but just to remind you ever goal the precise definition including age forest we don't said 1 contains an age borrowing he sustained major foreign workers In the momentum P Eagles miners bar by IDX same age a man down through the arbitrary where we put it by going over here I guess we would say that the change in society I Epsilon alright each board conjure up probably here in them just remind you want to keep your to keep you awake put image bars now and then every so often randomly OK where are they eigenvectors so Of the angular momentum the eigenvectors of being NDI did value so the
Eigen values of our values other possible outcomes of experiments and the eigenvectors states of the system which for which those outcomes of certain fighting cancer that we write the equation L on society the actions of the angular momentum is equal tour I get value caught Eigen value in its standard terms sought but but get out I would doing it I would have put Luelle here but somebody did this long in the past and happens to stand firm magnetic quantum number but that's a historical anachronism nevertheless I get value L and the let's look a little more of his picture but put v axis In actress comes out of a blackboard so we're talking about rotations about z-axis so we could call this busy component of the of the angular momentum it is a component of the angular momentum for the moment the suppress that right in a His value of that angular momentum so strictly speaking I get value of Z component of angular momentum was a matter which axis is and V in which actors text Europe take on our doesn't matter but its convention Boca had only following me eigenvectors lower we find the wave functions that correspond eigenvectors and for that we do as do the obvious thing we put in minors Our age Debye ID data that L acting on sigh of Fe that is equal to end the size of data that's a standard type of equation the derivative of something is proportional that's something else we could bring me I each borrowed the right side and that tells us that the eigenvectors with I get the value 0 it is equal to eat and data very similar to the case of ordinary momentum hopes will missing here I'm missing beach bar age but we go downstairs recourse if you divide by age you see the thing that comes in Emily age for now unlike the case of ordinary linear momentum with another constraint nor the other constraint the show the most know it purports Bella anyway medical state is that we they that is equal to 2 pie the wave functions should be exactly what it was when favor was equal to 0 because going around by 2 brings you back to the same point source is a sensible wave function that sensible way functions should be unchanged if you go around the circle but to pipe so that says that each on in old each bar times Tupaiidae 1st of all the wave function for fate equals 0 just 1 a fading equals 0 this is just 1 if I go around by 2 and so will be eaten by apartheid M all age forum and that also has to be equal to 1 What's the constraint strength is that end of age bar has to be an integer In order to make sure that the wave function comes back to itself a single value that has a definite value each point it's necessary but the Eigen values in the beat integer multiples of plants constant starts right back and it is equal to an integer now young and isn't integer times age bar but the usual notation usual definition is set in the equal to the integer just call Amis integer the angular movement of the Eigen value of the angular momentum is an integer times age that's the quantization of angular momentum in of H. Lawrence hereafter from for a time being I'll drop each part I would just say that the angular momentum must be equal to an integer angular momentum must be equaled when integer and that's fundamental fact acquire mechanics that 3rd we've known since the kind of news board OK now I have and the following question for you must think beta case what's opposes energy follows the energy depends on the angular momentum of course does the energy of the bigger the angular momentum and the bigger the energy particles would around the large angular momentum that has more energy than for a small angular momentum so we expect the angular momentum that media energy will depend on the angular momentum but must still be must be that the angular moment be energy our state with angular momentum aim not stallion value must be the same as the energy of a state with angular momentum minors what's the difference between a man miners well the only difference that may give positive angular momentum means a particles going around the circle counter-clockwise and negative angular momentum means going around clockwise there what many people think that the energy must be the same am the minus and course his money and to really think it does not have to be I will also raise my hand move it and that will get it but we catch fire not telling you anything about the direction of motion of a particle it's telling you which way you counted the data was positive this just as fit as I tell you I decided that a positive x-axis points that lady that does mean nothing about whether I'm
walking that way on walking this way it does it yet the definition of what you mean by positive and negative angular momentum but does not carry where the system has passed their regular annual momentum look at what you raise in the 2nd time you known example where a without the let it that docket cracked labor known example of the sin of a field which would lead To an asymmetry between positive and negative angular momentum regular velocity magnetic field left there were and magnetic field that's exactly what magnetic fields don't magnetic field for example I a magnetic field into a black will raise the energy a little bit maybe I have ever have backward raise the energy of a little bit of positive angular momentum and lower the energy a little bit of negative angular momentum and it will break the deed generously will break the generously between him and minus and that's the language which is another magnetic field the energy levels would be generous and so angular momentum Zero range were there and the all angular momentum 0 would not be degenerative only 1 of them angular momentum plus 1 minus 1 would be to generate momentum to win like this too would be D Jarrett in the presence of the magnetic field that generous is broken but that magnetic field does not violate in any way rotational invariance magnetic field as pointing into the blackboard it doesn't in any way violate the invariants with respect to retention and so we see that on symmetry is itself not enough not symmetry rotation symmetry by itself it is not enough to tell you that energy levels of degenerate Wyoming even talking about well but you need to if you 1 more symmetry 1 more symmetry is enough to tell you that the energy levels have to be generous symmetry very simple it's not continuous symmetry into discrete symmetry it's simply a discreet symmetry of reflection about 1 of the taxis reflection about axis is basically mirror reflection you have a horizontal India and any system above the mirror has a mirror images below the lira and if there's symmetry a not call Meris call reflections cemetery I'm going to call it symmetry and use the notation In described the symmetry the reason is I don't want to use Our from reflection because they don't want to confuse it with rotation and media began stands for reflection about an era and in particular we could have imagined the directors oriented along the way the x axis America's two-dimensional but this situation along the X Z plane but we are not the think about z we just have a rural the mirror image of former Ford draw the mirror image of that I don't think it can we get that right maybe not either and their image of this is but it's OK if there is not a mirror symmetry of a problem then a configuration like this at the BI don't what kabul building blocks of something the energy of this does not have to be the same as the energy of that there is the symmetry if there is no matter symmetry in the system but there Is there a cemetery it means that the description of this description of this are indistinguishable and they must have the same energy so lucky and symmetry 2 rotations symmetry and see what we buy from well we will certainly find that they energy levels Of the energy levels ER and must be the same as EU minus and wise act but think about in the upper half playing here let's think of a circular motion morning counter-clockwise what's the mirror image of that that's a circular motion mowing clockwise so the mirror image of 80 particle moving with positive angular momentum is a particle woman with negative angular momentum and if the world war system has such a mirror symmetry so such reflection symmetry that it means that the and energies of the same of the 2 states must be exactly the same they only have an example rotational symmetry which tells you that the energy levels depend on the energy in the were prepared that they depend on the angular momentum and the angular momentum is conserved be ingredient is the mirror symmetry which tells you that energy levels have to be be generate so that's a case in which 2 symmetries combined the tells you that there have to be be generate energy levels by a he tried we're sort of life each while changed might of and the that part of of it it seems it but sees that the site played let's say others will only racing and violence margin rate show that it Eddie ANX that moaned a function of failure is
not the same as a function of the data personable there's a function of data portability was concentrated over here is a corresponding function minus said because Soviet-era them up a also Ito the I say it was early to the I fate was not the same as economic science this is his 3rd them but that's right out of the school area this is the equation for the eigenvectors offside on eigenvectors of angular momentum Amal doesn't involve star Cy Young assault but acquire McCarrick swifter work away functions now probabilities that but there is no guarantee that no no no it doesn't it doesn't solve it through trickle what's necessary is have 2 cemeteries and 1 what condition that the 2 cemeteries don't commute with each other the symmetry operations of the 2 cemeteries don't commute with each other similar to prove that of symmetry don't commit with each other that implies generously not approved by Mr. Howell works in a couple of examples and the symmetries of like new commute with each other so let's Esq weather the rotational symmetry and reflection symmetry incidentally on the term mirror symmetry happens have another meaning in physics it's a very fancier turned mathematical terms that applies certain manifold and string theory don't get confused by 2 completely different concept I just use Mira tour represent reflection space yacht yes very interesting got good for it yet the U.S. symmetry of a magnetic field In version or magnetic field is a magnetic field pointing in the other direction if you take on the radio magnetic field pointing into the blackboard and you reflect the configuration about 80 . clean like this what you get is a configuration with a magnetic field coming out of a blackboard this is because magnetic field as an axial vector occurring you see it another way magnetic fields by car current lets us no matter what the magnetic field was there but instead imagine that the magnetic field is being made by current going this way the current going around globe like this it could be going around the solenoid we could be talking about a solenoid into the black bored with the current going Clark 1 that creates a magnetic field going into the blackboard all I have playing the mirror image of that there's a solenoid in which the current is going in the opposite direction and because the current is going in opposite directions and makes a magnetic field pointing out of a black board so yes Leon the Mary version or magnetic field is not the same magnetic field so I think you foresaw that that had to be the case the 1st of yup pig farm ask holiday as the mathematical with took the mathematics now of showing that reflection refer reflection was not continue With the rotation although that by showing that the generator of rotation doesn't commute with action of a reflection operator leader figure out what the reflects operated does the reflection operate will call a M From it takes any wave function and replaces it mission right arrow equals and replaces it by way of function of minors later reflection about the horizontal axis just takes played at the mine later so anyway function when you reflect it goes the size of my it be particularly if you apply it to eat about it will do in now let's consider that kind of data all 4 the mere symmetry transformation with angular momentum the angular momentum is just a version of our rotation operate of very small rotation let's consider b commutator for the commutator is telling us about is what happens if we do the toll operations in the opposite order a rotation about an axis and reflection process of reflection and rotation about the axis of same operations rotating about an axis in the reflecting is at as reflecting in rotating OKB and is obviously not all of obvious to you think about it but will work out by doing mathematical country and seeing what we get let's look at the commutator and would l we know what l here it is differentiates with respect later what does and build a reflex so we can figure out the commutator as we can figure out what does when it acts on a wave function act on a particular wave function act on a wave function Ito the RIA M later eigenvectors of the angular momentum and unwilling each far out of it we don't need and don't confuse begin with little lamb Little lamb and begin have nothing to do with each other this stance mirror this stems from magnetic quantum number OK but still work it out but 1st duo in the failed Times Ito the I What does it il do when it hits Ito the iron later it brings an end them right it just
brings an end so this gives us in the funds little and Peter eye and say that what does capital what when it hits including I am later it just changes assigned the favor incidentally wants this is just the number it doesn't matter which order began limit Peer services just equal will end times each of them minors I can say that the begin just reflected Stewart stood me up a and lead to the and they now opposite order 1st in acts and that gives us a L when it Max it changes the size of a there and gives us minors I am the lanterns ill hits each the minus science favor you know you have a good idea but it brings down a minors em by reflecting we've seen as a sign of the angular momentum so the angular momentum operator gives Simitis of what it came before it uses minors and he to reminders I am so be so it is that In an AL angular momentum and reflection operated don't commute with each other that the key to whether or not to symmetries on whether or not symmetries create generous use of energy levels we've seen in this case that 2 symmetries rotational symmetry together with reflection symmetry Cal that there would be generous ease of the energy levels that In an miners and have to have the same energy that's a pattern theorem but armed with a pair of would 0 right now the dispute any time on the Heisenberg delegates and fee for the problem is Alexine piano conserved right enough symmetries in themselves in this in the case of in the case of particle having a magnetic field the angular momentum rotation and reflection of Boston that both commuter the Hamiltonian both both the conservative extant pedia are typically in generally not a served if they worry the knee and so would be yes the Indians would be yes Wheeler study an example in some detail tonight with people get through it all that's well what's just the right now some form observations what I mean supposing you'll have to cemeteries nor is it means to have a symmetry but scored 2 cemeteries AT & and beat them for operators operators of act on the state that they could stay in the infinitesimal generators although they could stand for a full operation of the symmetry in let's say they refer to the generators take a case of where winner there are small versions of the year of the transformation that To say their symmetries means that both of them commuter the Hamiltonian it does not in itself say that a commutes would be and general here's an example where they didn't come they would be maybe not 2 possibilities both possibilities will lead to be generous use got books the 2 possibilities want possibility the commutator of a and B maybe a itself it may be be itself or maybe some linear combination of a and B that does not lead To new symmetry it just doesn't lead to a operators just leads to a linear combination of the old ones but let's suppose that leads to a genuinely new operators that cannot be written as a linear combination of ANB let's Causey OK 1st on the side of the are hardly see also commutes with each going but from the body in here why am I bring in there the commutator of Tova mission operators not permission violation as like the commutator extant P S if you want a mission operating he so let's suppose that seat is commutator of and B except for the next hour I see is our mission operator and only show you why it conserve the concerted trivial reasons let's think about the commutator of C with that a B a Each minors they a bit you're right a B is equal to see apart from a factor I so the commutator of C which is proportional that a B H minus AGB but now all we have to do is say Look we know that deep commute from age that's an assumption that means silicon will be to the other side wants it will be to the other side we know that 80 commutes with age that's also what assumptions original acre the other side and that of course is equal to 0 a and B both commuter the Hamiltonian then the commutator are 5 of 18 B also committed Hamiltonian defective product may be commuted Hamiltonian but the reason will consider commentators as sought the product of a and B commute with Hamiltonian certainly the commutator after I'm not so interested in the product because it's not generally nation its economy which is Russian so there's an example where given to cemeteries if they don't commute to generate a 3rd of a trillion and might be an independent lawmaker new things I want to have a and B and C you could play the game again commute see with a and B. Stewart you get you may get something that just reproduces are just a linear combination of the 3 you started with or you may get something no immediate thinking doing this and doing this and knowing there's until you don't get something new you might never you might get incident number of different symmetries that's rare that's unusual bazaar it will close eventually and you'll have some finite number Our symmetries which all independent of each other and which formed a kind of a closed system when you commuters new commuter generators but young are showed file it does end since I gave them
but eventually if you have a few symmetries start commuting Nevin commuting I'm among themselves being guaranteed that what you get is another surgery which may be just a trivial superposition of the of the original ones or maybe something no there is something new good but list and commuters Samoa eventually newly the run out of them were you want but I'm particularly interested in the situation where you run out and then you'll have always called a commutator algebra the commutator algebra includes the comic pages of all the symmetries and it includes the commutation relations with Hamiltonian which just say 0 when your situation whenever that happens to have a great deal of power over the system could be able say or a lot about the nature of the energy levels and generous use among men OK so that's that's the general always it was such a structure called the move a collection of symmetry operations are Group is a symmetry group is a symmetry group there there she has jet what worries the certainly be important for example we want to know from 1 state you could the photon go down to the other state can do it guarantees of the same we want classify the energy levels systems wine because respect Trask appears to me want to study the spectroscopy of of Adams spectroscopy this only want a study of the way take John excited states of page on pie on whatever same rules his father well with these things themselves is these I imagined were generators the generators algebra commutator algebra algebra we gently symmetries themselves which are built up from the generators by applying small transformations repeatedly bows are the group elements William mainly work would be would generators the generators have a lot of power the algebra the gap young it also now they're all they have is situation where you may have many symmetries but they commute with each other that can happen are then you don't get that you don't get anything terribly interesting you just have separate symmetries the commute with each other symmetries of that kind called at billions and billions means everything commutes with everything else when there are none 0 commentators between symmetry generators called on the the billions they still she and generator cemetery years if you take the symmetry operations and U.S. unitary symmetry operations and you consider the case other asymmetry operations very close to the identity and the case of rotations in the rotation by a very small angle and new right at than the former 1 I can never remember I think it's minus I G times Absolon genes called generator you could build up a few not generator that allows you to make a small rotation to keep doing small rotations repeatedly about a given access to eventually build up a whole rotation find a rotation so the generators and clean up basically all the interesting information about group well almost daily N C E no not start a B and C the yeah well maybe yes maybe now you might think BMC and they might not get it did DD Form might want you might get a Class B you have a plus B and it wouldn't be anything new we might get a because people back again still wouldn't be anything new but you have something no D which was not among you previous less then you have a genuinely new symmetry yet now that's not a question of the Dirac stories when food U.S. which I wanted to tell me what you think but you request of now and all the other is the collection of generators Ucon then they're still generators you can subtract then you even multiply them by ordinary numbers and you could commute them is what is it that's correct commuting is a problem that algebra and resisting is a collection of elements that you could and you could multiply by numbers and you can multiply so purely before multiply by each other in this algebra multiplication is commutation that's correct that's correct OK so we like an example down over like an example would likely pre-eminent most important example them back His angular momentum and rotation about different taxis as an example of what I'm talking about here we can consider rotation about the X axis which way except that we get rotation about the X axis rotation about the Y axis and think about you probably know their 2nd ever do summer Cartwright but you have to rotate the X-axis erupted about the Y axis of right amount would you get is a rotation about z-axis what's more if you wrote in a little bit about the X axis in a little bit about the axis not a little bit will about the war It's not the city as rotating a little bit about the y-axis 1st and then about the X axis don't commute with each other with a show will show that by studying the commutation relations of the angular momentum looked puzzled I know the best on but the In OK they don't commute to 2nd order in small numbers are that's enough that's enough but it grows the Peter remember these generators have the small numbers factored out of them and they don't commit they really talk it will example detail blue door right now the example of the 3 components of angular momentum 3 components of angular momentum form of the frequent rotations about any axis formal group but drop purposes now concentrating on groups the angular momentum generators former Dili algebra commutator algebra and simply means that the commentators aft components of the angular momenta give you other components of the angular momentum toward them their 1st let's talk a little bit about classical angular momentum from minutes just to understand the be corresponding classical concept on Docket reps
imagined a planetary orbit from my purpose now would be a circular orbit ajar milling about an axis and has an angular momentum which is perpendicular to the or be Elzy it has a certain energy he questioned a classical level is that energy levels be generous them meaning of that Is there on another orbit with exactly the the same energy of courses are and many other orbits with exactly the same energy but in particular the summit you can easily proved the same energy because of rotational invariance all you have to do is look take this about taxes what's rotated about the y-axis takers configuration rotated about the y axis entered no configuration end the angular momentum has changed hopes that's OK just where would deal like my picture record Cher's this is the wrong place To be coming from here there's Elzy and now there's a change in L. which picked up the next component fears of fresh picked up a net gets compartment and because of the rotation about the Y axis rotation about war boxes not only given us another configuration with the same energy but could you give us a configuration with a shift of the angular momentum and particular a shift of the x component of the angular momentum classically that's an indication that quantum mechanically the components of angular momentum from Camille a rotation by 1 Our rotation of a component of an angular momentum about different backs changed the angular momentum about the 3rd axis that has a great deal to do with commutation relations of rotation but will work it out detail so his example classical physics visitor classical level the combination of rotation and variants rotation variants of the combination of the different rotation variances about different taxis tells us unequivocally that there must be be be generously of your there must be orbits with the same energy our would rotate angular momentum that's that's a basic idea classically so let's go to acquire mechanically the 1st we have to do as understand the angular momentum generated better so what's comeback the angular momentum generated former previously I just said miners ID byte later but let's let's work them out not assuming that a particles moving them a circle but that a particle could be moving in 3 dimensions of factors new really have to be a particle but let it be a particle particle moving in 3 dimensions 1st before we do 3 dimensions votes the 2 dimensions particle moving XY plane not restricted to a circle and let's ask What's a generator of rotation now it's not ideal made maybe ID baby data but we can represent the terms of the x y coordinates 5 years the position of a particle let's say let's suppose we won't take the configuration rotating the configuration whatever happens to be means whether the particle is aware of the wave function of it is shifted so the particle point X Y and moved to another toward point what's the No . no point is given they all play was given by X Y and the No . is shifted by an amount built bags and why I'm assuming that the angle is small and let's call the angle a epsilon call angle Epsilon then what is dealt X Delta wine I'll leave it to you to prove this is a very simple exercises just simple geometry for small epsilon but calmly without Texas cost proportional someone might 1 if is positive and you rotate the change in X is negative so you gotta be miners and is proportional white but on my daughter y epsilon times X OK now let's consider a wave function sigh of X and Y what is a change in the way of function if you reconfigure the says the system by rotation sequel derivative of with respect to acts kind a change in X plus a derivative sigh with respect Hawaii times a change in white 1st shift away from but don't tax is minus epsilon times but what might Epsilon times y and this is a plus and so on times X that's the change in the wave function and of course must be by definition I guess it's L sub z site sell or at high Epsilon Phi Epsilon now it's much late leaving from follow the signs so I won't only the signs for you to work the figure at yourself but Our has formed what is decided by DX quantum mechanics I times PEX so this is another factor of sorry and this it is white times PEX matter which all lawyer I put down 1 PX year a commute PLY commuter P X X doesn't commute with PEX why doesn't commit would PYA but X commutes with PY ruckus this want this 1 is plus on tirade whose X P Y but if that is war crimes quite apart from site which is said I wouldn't try to track in any detail it's quite clear what the correspondence between his e component of the angular momentum the generator rotations and coordinates and momenta of a particle are Elzy but Mr. be identified with XPY wide and now I do have this is the right time and lost track of the signed but this is the correct size be seen that before
that's a component of our crossed P stay in the three-dimensional angular momentum are Cross page so now we see a new interpretation of our cross P not just the mechanical angular momentum from classical physics but now has the role of the thing which generates small rotations by acting on the wave what about the other components of angular momentum you could read them off just by cycling cycling through the X Y Z X was Y Y goes disease egos facts had gone through it X X coastal y y goes z minus G P Y and L X X Y Z P Y PCP Exs minus X P these of course you could confirmed by doing by imagining rotations about the other acts but despite not just by parallel reasoning this is what you get questioned for right please read I have 2 different notations busy I will Burma which or which runs a L L Y good thank you L 1 good could be OK reason that these are other symmetries for those if the system is rotationally invariant White well because rotating the system shouldn't change Hamiltonian so we expect we fully expect that if we have a system with rotational invariance then we should commute with a Hamiltonian typically they do they do they always do with the system has invariants on what kind of systems have rotationally invariant well under a very particular system is a particle moving in a central force field a particle moving a central force field will have applications and variants and therefore for that kind of system the country of L X with age equals 0 and in fact all 3 of them Elsa by 1 2 and 3 oratory Missouri they all commuter the Hamiltonian an orchestra of and this is just gold angular momentum conservation except the quantum our guys next question do they commute with each other suppose all we had was aleksandr Y that's all we know about Aleksandr Hawaii why we only knew about Aleksandr wives and mothers only Commerce about X wife laugh when we discover Elzy us the Elzy another symmetry mothers only told us about X and Y but they also told us about commuting generators of Lee groups of armed tells us work commute Alec's NLY and see what you get wet commuter Aleksander wives what we get OK I will you need to do we need to know the commutation relations between X Y and Z younger 1 he and the PEX 1 busy on the other hand the rule for mobile radio our Andy acts with was early that as always equal to 0 x itself as equals 0 x y X V is equal to 0 0 they commute Peters commute with each other P XPY disease or commute each other which they don't do the things that don't Camira are X with PEX and that's equal top body strictly each bond say will wiII a wire with PYE His i.e. frenzy with PC is I already Baron about X with P White commutes multiplication by exit the rift differentiation with respect Hawaii commute with each other and you could check back OK so that calculate the commutator of LX with a white caps commutator of white P Z minus Z PYA calmer Comerford commutator ZPX minors X PC now are not go through the rules for commentators your member dont about their berths by inspection Morrissey with cool in this side those commute with cool underside commutator this thing with this thing what on this side doesn't commute with something on the side while white commutes with me why commuter PX himself or but P Z does not continue with Zea so there's 2 things which starred would end of here Z doesn't come you with easy With Pezina doesn't only combinations if we were to expand this Our the only combinations which don't commute R P Z the girl with a Z and Z ZPX murders not pulled have here former my dorm commune Inc Bell X I will try white Pezina ZPY With LYA which is ZPX minus busy that from this charm over here you get a commutator of PC was EU Secunda caterer appease you with Chu about might my decisively Simitis and that should have PC in in your leftover with white times now wives are that gets rid of z with cheesy and that leaves over Hawaii and PEX from this would this year White PEX is a matter which already you write down 1 in PX not focuses minors on a white terms PX and in other which is commentators see with PC is again party and now we get RED XPY I believe yacht extra what extra PY from which is our friend plight times L 0 z too is an example of commuting generators and discovering new generation not a big surprise but nevertheless received before our eyes and so now we can write down the general list of commutation relations notice that by committing to angular momentum we got a 3rd
1 is only 3 His no way with giving it more than 3 generators are a sequence of overcome mutations and serving only algebra is just gonna be bill the algebra of 3 components of the angular momentum but what's found 1 says that no acts with the hills of I Elzy and then the trick is always the same just cycle L L O equals right yellow Of course a fellow brackets calmer X Y Z why Z X zee Z X white buzzer basic commutation relations of angular momentum if you continue commuting you won't get anything so we discovered is a genuinely algebra yacht would chill with all you out your drove that is not really a come yeah I suppose we do but DIP we prove that but see what we do not Lula Mae not bans that right now are them that gets me off track it should be a glitz a system back to it should be very easy adorned Japan or get my head that confused does clusters component of the angular momentum is not likely to be linear combination of X of white components but yes you're right you're right we do have the obligation of showing that nosy arms it is not you know what I think can probably show that Elzy from these commutation relations I believe we could show that Elzy does not commute With any linear combination of relics of a life our if els e was a linear combination of why would commute without linear combination I think would prove it thought the at wasn't it was mean that I like I guess they are components they'll put perpendicular component of three-dimensional vector or chorus peak but I want use that I think the right thing to do is exactly is Jones said is to directly prove algebraically that bills ease novel a combination of the others think it would prove that gap was sent only algebra is an algebra where the rule for multiplications commutation means that steps but don't worry about the name of the name is just did just that it's a collection of objects whose commutation relations closed a collection of objects when you keep commuting you don't get the 1 that's called early algebra here there on that system Maine we of course was a mathematician near their was algebra mathematician AII might lend them was a beer In biggie was now whatever work was set up with it is an within but it wasn't I have a feeling his name was Al Jo beer the your reading of the video remember reading OK so where are we we have this nice collection of algebraic relations including for 3 more voters right there was 1 more time if there are any new component of L with Hamiltonian is equal to 0 which means the 3 components of angular momentum are conserved appear what do we do with that well thought visited the time to remember what we learned about the harmonic oscillator by having a similar kind of algebraic structures a similar kind of algebraic structure that we had worked commutation relations of creation operators annihilation operators and the number operator we had closed commutation relations are very similar kind not exactly the kind of close commutation relations and they will quite powerful we deduce the entire
spectrum of the harmonic oscillator we did from constructing creation and annihilation operators who was much easier than solving Schroedinger equation from those same thing here construct creation and annihilation operators but the creation of annihilation operators I'm not going to change the energy for the change is Zeke component of angular momentum no while I picked busy component of angular momentum that's arbitrary member in quantum mechanics if you want actually work with things you usually have the pick up basis you usually have to pay it is beyond a preferred set of operators which you regard as as defining a basis often the doesn't matter as the X basis of VAX representation of representation in quantum mechanics of a particle is position representation of momentum representation in studying Spain's we have received Mazzini representation and racks representation and so forth a really doesn't matter for the purposes Of the invariant physics which bases to use but it's common to pick 1 work with it and keep it could do the same thing here will pick out Z component of angular momentum ours the focus on there's nothing special about it we could pick the extra we could pick the goods useful to pick busy component no more useful Aleksandr Y. but let's do it OK go work with Izzy component of angular momentum shares and eigenvectors was a component of a new momentum and its eigenvectors and see what we can find out about the entire around Spector OK if els e has eigenvectors permission operator a harmony and has will find out as we go along and let's call the Eigen value lousy little and it's the same Eigen value of saying that patient used in 2 dimensions in dimensions where was all leaders E. component of angular momentum I call the Eigen values M. Carpenter new to do that again if a historical notation they had to deal with putting Adams and magnetic fields you put that magnetic field and you studied spectroscopy and arbitrarily busy direction was chosen for the direction magnetic field and the angular momentum about z-axis was called the magnetic quantum number but the drop purposes now it is just an I value of else of GE misses the equation that says that the eigenvectors him is isn't eigenvectors subsidies would I get value will let so redundant notation I've used the same location for the eigenvectors and the Eigen value and here's were mathematicians go crazy because they don't want to use the same the patient 2 things if but we're very focus of effort that wanna find out is what other possible values of and the possible values of him will be exactly the same as possible wagon values of exit of why because nothing special about the nevertheless see what we can find out now on the case of the harmonic oscillator we figured stuff out by inventing creation and annihilation operators or operators which raised and lowered the energy now invent operators which raise and lower they raise and all the magnetic quantum number and how do you do it you do it way very similar to a harmonic oscillator young hers but fast we did it the quality we think we know that lot less but Boris there is yeah a that will read and we never did the harmonic oscillator OK you in that case will come back to the harmonic oscillator so tell you how much this is like the harmonic oscillator I will tell you how much the harmonic oscillator was like angular momentum who left this is self-contained doesn't require a Samoa at us set worse said Jan my I've forgotten to it's an book but positively get mad if that like for picking not worry about the harmonic oscillator much not complicate this so it think battles is a little bit special not physically but mathematically for our purposes and invent operators L. X
classes I wife and also LOX miners wide when called L plus in minus what wanna figure out is the commutation relations of L plus in L. minus with Els for the moment I don't really care too much about the commutator no plus with miners have played cards Elzy knocked down start from put what I want is the commutation relations of the plus with minus so heavily get them well Armed with forget get them from the last tool of these equations but they write them I want to have a record L X would Elzy Mrs. 1 over here I've she'd be Order of exon disease that minors on 8 0 White hurt right jock that's the story or here I change the order of X that changes the sign and the other 1 commentator of L wire with sales and that use for 0 I L Y we have to do a little work his L with Milzie multiplied by party and we get Miners LX and now a disconnect and and subtract these 2 equations begins attracting them will give us the commutation relations between El plus no miners on the 1 hand and elves my OK we them subtracted free and then we get L class commuted with lazy and if we subtract them we get L minus we do the summation on this side also owned and he is or we get on the right of our we don't need to her of go labor at time a R L passed with 0 z is minus L. class and the comet data are of miners with Elzy Israel might not minus 0 minus plus or minus 0 . both the 2 basic commutation relations that follow from these guys understand this is very abstract have anybody know to do this while on remaining
intruded its public demand that I would remind her you like the commute the that year that I can't tell you exactly why you did this but you don't want to do it is they who are located at the very clever so this is just another way of writing we algebra it's probably true that commutator helpless with minus Z Our people maybe twice sells before but we will use it now knowledge approved the following very interesting will theory supposing we have and eigenvectors of aim of ills but only have or we found 1 somehow none find another 1 with a different ideas value I think we could do it by up 1 of these equations which 1 we do help last looks to Alplaus with take this equation fell plants and multiply it by the eigenvectors in the school eigenvectors and so let's Woodward work it out help lost tell my nest sorry l plus Elzy miners Elzy plus that's the kind of Qaeda acting on the leg that and is equal the miners L plots acting on a vector areas OK so utterly normal let's take this 1st turn he well OK let's take his 1st turn here a laugh What does it Elzy do when it hits and multiplies wagon as because and is and eigenvectors bills it so the 1st turn becomes in Times l plus and promote take L plus in an animal put them in a red circle I want to have a figure was a unit now let's end this overthrew outside here we take this Karim and move it to the right and the thing this 1 with little left so let's take this 1 and over or August plus help plus and that's also exactly the same thing that appears in the circle so where we get we get last 1 times object in the red circle as just draw a red circle where that's the left-hand side with this term transposed little left now let's try and for the other turn to the right that equal To Elzy kind of Royal plus again exactly missing in the red circle trimaran circles a certain kept record so what is a site this says that is another eigenvectors of those Adidas equations and eigenvectors equation for trails Elzy acting on the red eigenvectors gives us end plus 1 kind of argued that was them easily discovered another eigenvectors of LCD where the Eigen value and plus 1 we can call the thing in the red bracket here we can just call it the eigenvectors and plus 1 we found that Shrek for taking any eigenvectors of LCD and promoting it lifting up 1 unit and increasing the eigenvalue by 1 year what about if we did the same thing with Elm minors if we did the same thing with minus we would begin discover another eigenvectors within 1 lower unit so we've used a simple algebra of commentators to generate a suspect charmer spectrum of values of bills and more more separated by an integer are separated by integers will discover the following fact that if we plot on the vertical axis year of possible values the values if too much Elzy Elzy the possible values if we find 1 and some value of end of what 1st of all let's armed but for some of us lace integers Darren 0 0 0 1 2 0 0 3 0 0 minus 1 minus 2 minus 3 let's suppose we found an eigenvectors Eigen value of and nothing yet his told us must be an integer so it could be as Kabila here at Kobe a three-quarters of abuse 7 8 ca be Pike quarters be have it could be anywhere is notorious for over here they push but it was reduced if I could get find it has to be a yacht is New York right you're right except quite but yeah we could go back to what I said earlier and use that I don't want to I don't want to because I want to show you another argument for it now the other argument will not prove its integers proved something a little bit looser and looser fat is in fact the correct but also a 5 star of the 1st thing is we know that there's gotta be another 1 above it 1 you knew about and alone when you the others want exception to this statement that will only apply L plus we will necessarily get eigenvectors would warn you about what see other alternative possibility gives you just plain 0 no vector so this could terminate conceivably could terminate I wish I were green and it could generate somewheres terminates here and by the same logic it could terminate from below 0 if L miners killed the vector so we will generates a family which may or may not also terminate somewhere now 1st thing which is true Oh is is a relay if it terminates if it terminates them as a relationship about where it terminates from above and below and the reason is very simple and easy I get value of Els remember we have rotation variant suppose I use my location and various to completely rotate by a 180 degrees of our rotate 180 degrees then Elzy becomes might ourselves if it is really rotational symmetry that must mean the spectrum of the possible eigenvalues Elzy must be the same as the eigenvalues of my sells it this is looking at the same system except the standing on your head was rotation variants you should get exactly the same spectrum so from that you could conclude that this adopt and this red dot have to be symmetrically placed there's only 2 possibilities if the red symmetrically placed and these every on and these are separated by pitchers there was only 2 possibilities the 1st possibility is that the red dot is an integer which cost us is integer n n 0 1 2 0 0 3 1 2 3 Sarah here symmetrically
replaces symmetrically placed about the origin and separated by integers only once 1 possible spectrum 1 2 3 0 0 1 2 3 if it victory terminates is another possibility and the other possibilities still is consistent with the symmetric spectrum separation by integers and that the half integer option is Oregon 1 whatever start here shift by integer shift by an integer shift by an integer say here 0 1 2 3 1 2 3 and also symmetrically placed about the origin In this spectrum over here if it's physically reasonable the values Of the angular momentum Uzi component the angular momentum are half integers digital 3 5 Arabs might have my history have my 5 ending somewheres more the other possibilities in respect now you mentioned before that there are rates are approved that the spectrum as integer did somebody mentioned such good that is correct for orbital angular momentum or below angular momentum means or across P means wave functions of particles that depend on position and how we proved that proved that would prove that by staying away from China's to be single-valued when you around always around OK so high on the face of it it would seem this is the only physical possibility in fact this possibility here is also Rio and it corresponds to spin angular momentum but for the time being was just get rid of it will come back to bite as Mac give a state like this we can generate a few more mainly that terminate but let's assume a terminates we found in multiple it of state meanwhile multiply the states which are in some way related to each other In which have have the same energy why they have to have the same energy but prove they have the same energy and they will quit that'll be enough for tonight but drew 1 more thing they have to have the same energy suppose that we found the state and then hasn't energy equal to E times and eigenvectors the energy it has L sub he Quebec has energy now let's go and plus 1 plus 1 is l plus kind and that's a plus 1 and let's ask what its energy is so where we do we apply the Hamiltonian correct and see what we get but the Hamiltonian Camille assumed that the Hamiltonian commutes with all the components of the angular momentum so that means we come right that this is the same as pope last H plunge and but 8 times and is just these columns was is equations set it says H plus 1 0 plus Sunday and is equal to E times and plus 1 really do the same thing with a minus 1 that tells us that if we find an eigenvectors of the angular busy component of angular momentum which happens also to be an eigenvectors of energy by raising and lowering until we run out a possibility until we run out until we hit the end points we create states of exactly the same energy this is the idea of a generous following from Symmetry when symmetries don't commute with each other we talked about the beginning with symmetries don't commute with each other the elves in this case it tells us that there aren't too generous and be generous cities our states of equal energy and in this case the states of equal and a geologist but found by going up and down ladder persuading state young there there've very closely related to the idea of taking a stake in rotating it yes that for example you could ask the question What are the eigenvectors of Ilsa backs the eigenvectors of Elsa backs are not individual 1 of these but linear combinations of right linear combinations of these are eigenvectors of acts Hawaii and it In that sense for finding here is zone essentially the eigenvectors rotated the rotated versions of the him and that's uh kind of obvious because when you apply in a a generating rotation so this is a quantum analog of saying that take in orbit and rotated about taxes you'll find in the war but with different angular momentum but with the same energy several different Zeke component of the angular momentum exactly what's gone on here rotating the configuration is creating states of different Zeke employed different Zeke component of the angular momentum but with exactly the the same energy that attacked by applying the or the rotation operators now it's convenient to use this particular combinations plus or minus arm just because Dirac figure out how do it this way this is on the trick of using raising and lowering operators in this occurs all the time in quantum mechanics when every you find an algebra like this which closes the commutation that's a signal that the review should be looking for a creation Oregon our raising and lowering operators and using the algebra sort out the spectrum of the system the spectrum of eigenvalues about that'll give 0 my definition that's a definition of the top level may not be a top image got a minimal the it was by the way you know plus give 0 by what but right now this could all be Maine concrete by specific wave functions a this happens I didn't want to write down complicated regular way function that will want possibilities you never run out of anything that would correspond to a basically infinite angular momentum state states of incident angular infinite capacity but typically real angular momentum states of atoms and so forth in some orders from work out more about angular momentum a little more and they will apply it to Adams and understand a little bit about comic spectra blew a bit about the spectrum of the hydrogen atom bomb this is where the generous sees or some of the degenerative voted for all of these exactly generous use of hydrogen and come from all of these equality of men and these states orbital go there of the various although quantum numbers over of the hydrogen atom was more was more but there choker for more please
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Titel Advanced Quantum Mechanics | Lecture 2
Serientitel Lecture Collection | Advanced Quantum Mechanics
Teil 2
Anzahl der Teile 7
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
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.
DOI 10.5446/15097
Herausgeber Stanford University
Erscheinungsjahr 2013
Sprache Englisch

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Fachgebiet Physik
Abstract (September 30, 2013) Leonard Susskind presents an example of rotational symmetry and derives the angular momentum operator as the generator of this symmetry. He then discusses symmetry groups and Lie algebras, and shows how these concepts require that magnetic quantum numbers - i.e. spin - must have whole- or half-integer values.

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