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Advanced Quantum Mechanics | Lecture 4

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the fish Stanford University a harmonic
oscillator let's just very briefly review of the harmonic oscillator because their come up time and time again in what happens from here on in as a set from my Opel oscillators are ubiquitous that although the place at any time you have a system which has an equilibrium state it could classical or to be quite chemical but there but classical in particular it has an equilibrium state displays From equilibrium generally speaking oscillate may oscillate with more than 1 frequency for example of violin strength can oscillate in the fundamental frequency can the 1st come on the 2nd harmonic ship plucking on ways you could make a some nasty noises hasn't sugar or hours many times the nastiness because multiple time user clashing with each other well that's probably quite right but the art nevertheless the oscillations of Ireland's collection of oscillators different frequencies but more important To us harm the oscillators which comprises oscillations of electromagnetic field and other fields will come out and do a little bit of quantum elected dynamics just a little bit but basically if you have radiation for example in a year and a cavity Oregon we've got something like that then the radiation electric and magnetic field oscillate with different frequencies begin what will be the equilibrium configuration of the radiation field mice start with something which has an equilibrium configuration What is the equilibrium configuration of the electromagnetic field the answer is no electromagnetic field the vacuum and these space with no electromagnetic oscillations and no electromagnetic oscillations in other words just heard empty space with no electric and magnetic fields is the equilibrium position of the electric and magnetic fields of if you give the system a knock by for example while a standing in some microwaves or something the system will stop oscillate and might be like the violence train it can oscillate in all of various frequencies and those oscillations column called folk on August quite right the oscillations of time or describe collections of 4 four-part of come but not tonight OK so harmonic oscillators terribly important in many ways incidentally sound waves are also oscillations sound wave moving through a crystal again the collection of oscillations you could think of it wrongly as each Christo oscillating independently of every other workers go so each Adam oscillating independently of bad but that's not really the way it works each Adam exerts a force on the other Adam and Sony displace 1 gives Notimex 1 waves go through the crystals the waves again are described by harmonic oscillators end the quanta of these oscillations Chronicle folk dance the call tho not as examples visit things were studied but there through prepare us also back we studied a bit the harmonic oscillator and I want to expand on the date remind you of what the rules aren't we started with a harmonic oscillator with a Hamiltonian which I rode as peace where all the twice the man and I set the Cooper 1 just simple you could go back and do it with advanced not equal to 1 I believe in our own not book we do with the general mess we do it young workers four-star general rest and Vin Plus Omega square best spring constant but also happens that May is the frequency of the oscillator X squared all the and we constructed creation and annihilation operators a plus or minus which happened to have been momentum plus I Omega acts now notice not finished yet but August if I get classically if I would take Foster mind if I want to take the tool operators a plus and minus and multiply them together and then divide by 2 will I would construct the Hamiltonian combat minute but in order to be consistent with the these notations of ancient 0 per square root told or may get here and those are the a pluses and minuses and if you work out classically or quantum mechanically Hamiltonian in terms of a plus or minus user-defined all 1 1 1 more quantity 1 more
quantity is called in In his equal pay a plus a might as simple as that and the last time we proved that the spectrum of eigenvalues of In were just the integers starting at 0 we showed that a plus and minus or create raising and lowering operators which raised and lowered the eigenvalues of services are operator always Eigen values are 1 0 0 stop start was 0 no negative 1 0 0 1 2 3 Bob Bob Barker finally got finally but more over the Hamiltonian is equal to or may now I love I have left out each bar and leave it out but don't just show you where it goes out each part of the Hamiltonian is just Omega times in now no way was I come from if you multiply 8 plus times a minus clearly you get something which looks like this was through downstairs this show and there's an Omega downstairs that's why I have to compensate would make it easier for me but quantum mechanically is a little correction and the correction comes because he's because a plus and a minus the from you and that gave us in plus 1 half the 1 here has called the ground-state energy of the back have also feel free to call the vacuum energy if it there is a simple harmonic oscillator we quality of Zero Point energy or the ground state urging that consequence I'll be uncertainty principle but you can't have both P N X simultaneously being 0 and since the Hamiltonian balls please squared past X squared is no way they can cancel each other was always going to be a residual bit of ground-state energy there you'll always very much if you just crossed out forget it because the inactive constant in the energy of the system really doesn't change its equations of motion so you could ignore it but it is there were some purposes but what I want to do was to show you and l Yao was 1 of us 1 other important points that L our equations that with data showing that they pluses and minuses were raising and lowering operators who was the commutation relations of the 8 closer name might Barak's rulers have 2 things commuter ahead sometimes a cost advantage of the becoming commentators are a minus with a plus just global warning had I not put the square root of 2 0 made is the commutator wooden something a little messier and so really the reason for dividing by square 2 maybe here was to get rid of anything on the right side the vibe away this commutation relations together with the definition of tells you that in is quantized in integers and the connection with the Hamiltonian such Obama can't Audi energy levels these yet like wiped state white after 0 0 0 1 8 0 0 is included OK William gaffe offered for the goal is to demonstrate explicitly what the ground-state state is that there is a ground state more Fung ground state but actually solving the Schroedinger equation policy that definitely is a state with 0 on what is a wave function a wave function in this context let's just over with quick parallel to this very tricky and clever operator of structure he lets right down the wave functions description of the same stated that is are replaced by a wave functions which are functions of position if we're talking about energy Icahn's states we could label the wave functions with a little it down here indicating which integer they correspond to the ground state for example will be size 0 1 of the 1st excited state would be sigh 1 and so forth so state vectors become wave functions operators X operators acts we can't just multiplication invite the cordon X I'm trying I'm trying to remember his capital letters for operators and lowercase letters for around just numbers lot case OK the
momentum His replaced by minors I be by just to respect X but again
an 8 foreign therefrom or keep track of each Boris always don't keep track of each bars His work hours right here and that's the only place the only place where you have to modify I guess maybe all sold here we want the board downstairs but Donovan Bailey you could always figured out with each bars go by using dimensional analysis using dimensional analysis and remembering units of age board will tell you uniquely wet but th borders OK so there's momentum and our let's check and see if there is a solution of the equations 1890 on society is equal to 0 and words is a state in I candidacy a question faster reviled limiting the media and your question quickly without combat for calculating armed we know there has to be a state which is the bottom state we kinky going down down down for the simple reason that the Hamiltonian as positive for peace square what sex square square pass on cortex and operate alike that has no negative I just as classically it in a positive 0 . mechanical you also can be negative and has got to be a bottom is gonna be a Florida Power of our energy levels OK how clearly how floor that can have a floor is if the bottom state but close 0 score gramm state is annihilated by be annihilated means that you get Zero when you hit it with a minor it did get 0 whatever you got on the right hand side would be a state of war Nunavut lower energy still state with 1 yuan lower way for that happened is for this to just to give 0 now once we know that a minus on 0 gives Iran then we can look at it again so let's look at end now what end do when it hits the bottom state well it is by definition a plus a minus on old boy already know that a minus on 0 0 give 0 services 0 eigenvectors game with 0 eigenvectors so there must be an idea of the bottom state has have 0 value of it that's an abstract argument but not so it does 0 is a theory that you're 0 far right side and this 0 here this state and this up to different things right that's the eigenvectors normalized 1 and it's not Zero sense it's just that it's being 0 I call if I called my friend are 0 that would mean I wasn't here with did exist been unpleasant . 0 it yet you connect you can write 0 as can but you have done the Davidson of this 0 this is your does all classes that they 0 times anything 0 super is danger of that the size of the daily here just as they know what you must not Kohut respect whatever being you give it to gotta give it time for bitter brew tradition is just not the riders is a vector but just call a 0 and the abstract theory of vector spaces you introduce a vector called 0 and so special was the only 1 that doesn't have a kept silent but as you say it is just another vector forever special 1 now OK the bottom of the tower the bottom of the year on the bottom of the energy spectrum has to be energy each borrower deprives one-half of equal to 0 begin equal to 0 also but let's see if we can see how following King goal in calculating although we've functions going calculate the wave functions themselves now you might ask I can write assuring a to write equation I just use the P E in the cop equation I used Dubai DX time independent Schroedinger their equation the right hand side just the energy eigenvalue kind society question is really true that you can only solve that equation in fact could write down quick what's right down the equation kind independence from here this time conjuring operation might just be by X squared ordinary derivative minus 1 half the 1 a offers the hair from former Hamiltonian there Our squared is Dedie 2nd by D X squared Silex plus the potential energy plus Omega squared x squared times Siurek if this is the kind independent Schering require you know what I want to do is to solve the equations that this is equal to let's call it easy for the Eigen value for energy time sigh of acts that same as writing he squared plus Omega square neck squared on a vector is equal to eat times and OK now really about differential equations you know this is a 2nd order differential equations you can always solve it any value of the energy here but but will happen for most values of the energy is the wave functions will not go to 0 at infinity even worse will exponentially explode and get big it won't be normalized it will not have a finite probability under it there's no way to normalize it if you try to normalize it it'll just be 0 because you may not be the integral on the square of the wave function will be infinite figurative that infinitely you'll have to divide after the abide by it and away from shore which B 0 everywhere so sold the rule is 1 will roll over talk about this last quarter of three-quarters of Aurora was is that what you mean by an honest vector here is 1 whose square In the growth was good the square of which is square-rigged available in the growing size star sign equals 1 OK there were some exceptions to this particle moving on an infinite aligned sometimes we introduce momentum states in positions that states which are a little weird in this way but if high bar we would like to public probability to be finite not exponentially exploded so this is 1 additional restrictions and its win the wave functions are restricted in this manner when they really former honest good vector space then that's what the rest of the supplies here only applies if were interested in the square in the global wave functions which we are
themselves the point is the only right that only right down near the Schroedinger equation there are certain definite values of energy which will have square in the global wave function as those are the true honest I function our if the wave function blows up in goes to infinity we just thrown away in space OK let's see if we can find signed not the ground-state wave functions we could write the original Schroedinger equation for it but we can be smarter and that instead of writing a chance I equals side with smarter equations simpler equations that use a fact that we know but that annihilation upward Lauren operator Excuse me the Loring operator on the ground state and the course I'm not just nor do 0 OK what is 18 miners 1890 s for divorce arms right a minus is proportional to the Pete miners Omega x now this some factors in the denominator there but we don't care about them because we're just going to write that this time sigh x equals 0 numerical factors like square roots or maker they don't matter in equation is multiplied through by them and they don't remember what Peters P is minors I by DX might Our Debye D X minors I Omega x prime side of x equals 0 now we have a much simpler equation than we had before doesn't even have any 2nd derivatives in that it only has a 1st derivatives silicon trader's Solvay on the track for solving an equation of motion might be limit in order trick as there is only a finite number of tricks in solving differential equations is a handful of tricks what what some Miller's only 1 variable we are separate that's that but get 1 very standard trick linear equations like this is a right beyond Mr. right away from exponential of something else now this is just a matter of experience I can't tell you how do we get they guessing because when all by DES so these days says it all the the time that it a 5 so this happens to be the kind of equation that use solved but you could solve many ways but there were a way of solving it is just the right side of X is equal to eat some effort acts and try to figure out what taxes so it's drive on 1st of all of course we can get rid of the IEC high I teII factor out furthermore we get rid of them minus sign minus signings and know what derivative of of acts the calculated derivative of sigh X we simply differentiate and refined what's courtside climbed which means the derivative is equal to half prime derivative of death times he'd be Everex to differentiate an exponential you just differentiate the thing that the the argument of the exponentially and multiplied again by half of acts into the effort heatedly effervesce OK that's sigh prior to this year and then we a enter wrapped so we want to get about plus Omega thanks sigh of X which means going into this plus all may get X Ito yes sex bias Michael were saying that Peter the FOX factors out flicking a of it and that's where the trip that's where a trick was on her 1st place available this year and our equation is just FTX plus Omega x equal 0 that's it equal 0 but the FTX came from here force Omega exit both multiplied side so we Garuda equation is is really easy to solve a function who's derivative is a linear failings whatever from Cher's quadratic thing so solutions quite clearly going to be proportional back squares and fact it's gonna be half of X is equal to 1 half Omega x squared plus a constant for the constant in there but the constant and that's all my might bring that your mind this 1 have Omega x square that tells us that sigh of X is equal to eat them a minus 1 half Omega x squared What about the constant well we can edit but that just puts a numerical constant from the exponential and I haven't bothered worrying too much about normalizing a marketer bother worker can't numerical constant out front is there it's going on data over square or maybe in it but it's not terribly interesting for our purposes this it is a sign off X and notice that it goes 0 0 very very fast with acts leak of a mind X squared it's a gal CNN bell-shaped curve that goes to 0 as each of them minus 1 half Omega x squared and is very very squaring the global square-rigged integrated it's extremely convergence and so it succeeded and for now we can go back and check the original Schroedinger equation is satisfied another words we could plug this wave functions into the Schroedinger Hamiltonian up above and calculate with the energy eigenvalue viewers already know what it is it's gonna be all maker times one-half we know that from all the algebraic trucks over here so we start this function back into the strutting around into this Schroedinger equation we would discover that he is equal to one-half just putting it in plodding union and Doran it so we know the ground-state wave functions we have want eigenvectors of the energy we see what's physical properties are concentrated near the
origin and very small because Agustine functions very smooth its concentrated near the origin it does look you might expect a ground-state wave function of a harmonic oscillator to do it the origin sits very close the origin or leased by close now I mean that is concentrated near the origin and gunned gap whips what about the 1st excited state of below the 1st excited state for you injury a Thomas trivial and then you can go home and have a lot of fun calculating next 50 states hurts so my Lewis far but honey daily he the head of the how you do it it's more interesting than what you get well on both interesting but you do it by saying Look I don't the 1st excited state is abstractly abstractly the 1st excited state is equal to any plus times the ground state I know what the ground state is an annoying classes so let's just applying a plaster the ground-state use the ground state a plus his people sigh Omega x and that will tell us for the 1st excited state rights or from a wave function assigned 1 of X is equal proportional to O P Plus I'll make X now he is minus RIE Debye D X plus I Omega x on each of miners 1 half Omega x squared so Soweto was calculated derivative but it all I really did its irony did this week where opposite side we did it with any money to give 0 if I don't live the opposite signed a did 0 0 that means if I do it would be signed Kia I'm just going it twice been so that I would get go back I did this operation on here and have 0 I swear solve the equation at this blows this actor on this gives euro must mean that this term in this term give the same thing apart from a sign for wants another that that I know where would you change the sign of leader over here and setting having cancel orders double the answer so this is just going to be twice my Omega that's not so interesting it's just it's just a numerical factor happens to be imaginary but who cares but as X times to remind 1 Omega x squared so structurally it's different that's not just a gal seen function sagacity and functioned kind X was at like Fergus again it's very small far away the fact that doesn't X here doesn't make it big far away because totally overwhelmed by the the my sex quartered yet for it negative not well it's twice imaginary number times amigo Our but whatever it does on 1 side has the opposite side of the site it's an odd function on meaning it changes signed with arose the origins right so website so that would make it negative on the side of the found and incomes up 0 0 is odd mediators say that when you reflect it changes signed its but In isometric you would say it and die that the probability which is the squalor of theirs is symmetric but most of the probability goes 0 OT origin it here and the 1st excited state of a harmonic oscillator probability that you find the oscillator at the origin 0 interest in fact Our that's a 1st excited state it has the ground-state had no no no means that the wave function zero-sum orders the ground-state had known of the 1st excited state has won each time you are actor with 8 plus you could you could just spent a little bit of time with casino works each time you act with 8 plus you make Ohio order polynomial in front DeVito of makers each time you hit it with another 8 plus it'll give you X glared back squared plus another terms XQO after while each time you go higher and higher order polynomial and each time your a unknown does next wave functions spin of the year yard the 2nd excited state of books as do 31 has 3 knowledge and a few white dealt the summer heat it looks something like each time your actually plus you warned that pushing away from out more for example 0 of the ground state was concentrated nearly at the center of factor of it acts here the push it out a little bit wiser push it out because X is bigger far away than his nearby and saw pushes AFP you've done 70 factors of maybe 24 or whatever you'll find that the wave function was pushed out of the wings small May the origin oscillating and and plop grows 0 is symmetrical and he is a magic Grant isometric as up to get more more eagles and the whole thing gets pushed out further and further what'll legal what the implication of the wheels Wales mean momentum what's the implications of it being far away from the origin of potential energy so as you keep exciting if you create more and more potential energy by pushing the wave function how where exporters large but at the same time you create more and more momentum as that oscillators wings of origin of what's happening is the oscillator trying to do with classical physics tells Europe below that the higher the energy the more likely it is to be found far away but also may be found close and found close have a high momentum so that's why it can stop oscillate very quickly the origin and more slowly far away but that's the basic physics of it from drew well it should also weighed more rapidly near the center and less rapidly far away just indicating a fact swings so the origin of course has more momentum OK that's so those are the 1st to wave
functions it seems like a very large course was the whole told house that would use correspond with that very large it looks like classes they thought about it In fact but you have to do to see the classical motion is you have them take wave packets you have the superposed many energy levels for example if you take the ground states follow the grounds that's a nice smooth wave packet now displaced at this place it off to the side he yeah it's no longer the ground-state in fact not even a member jargon say altogether it does have an average energy Lemelle take the time deep ended Schroedinger equation the time deep End shoring or equation has every energy level oscillate with different frequency the result is that this wave functions not time independent and what happens it starts swing back and forth itself swing back and forth as a is near the center gets lots and lots of wheels but stays of the same general shape wiggle comes out the side after a while license would begin swings back and forth and back and forth looks very much where classical oscillator the effect of piety energy the more it looks at the classical oscillator our just because the higher energy in the bigger the swings the smaller of the ratio of the uncertainty of the size of the swing when the swing is very big dominates the spatial structure of it OK that's that's the basic physics of the harmonic oscillator Yukon work at the sector level 3rd level a target of 4th level you won't receive any more but the straightforward I at us said yes change she hit you uh yes no I don't think there will I don't think so I think it that is a constant Tamils from your young that's not it's not just X squared and young recruits was overdue uses the doors of a young . I'll tell you where it's theory was nowhere to you in Estonia was Sonja it's assumed in a fact that's a plus and a minus Our Home Mission conjugates of each other if you would take a wider class of wave functions that were not squaring the verbal then a plus and minus would not b permission contradict each other that will their Medusa at 3rd place so that a His would grow if he our house and here there's a or at equity and bad but sometimes you encounter a solvable example Warner solvable I can usually be solved by these kind of tricks but there are very rare in the space of possibilities and by some 40 of accident the harmonic oscillator 1 of them and the hydrogen atom or be idealized hydrogen atoms nonrelativistic hydrogen atom would appoint nucleus happens to be 1 of the solvable examples OK now I have my notes to other subjects for tonight Armed with the idea of were discussing Adams but forever but Morgan Adams discussing everything there is about quantum mechanics and field theory things Spain and bows on firmly on different difference between all what bows answer for me are I have well I guess of law according to my to my notes here I'll follow my notes I have been the 1st and then bars on him so much talk about have this spin of electron spin of the proton which happens to be have been which means that the speed of the angular momentum of an electron at rest arrested has no orbital angular momentum has no are crossed P lawyer has been suspended as a kind of angular momentum that patch show particles it's your choice Woody wanted think of it is just an abstract concept but would be worth think of it as a tiny little thing which is literally spinning spending about Mexico's Isaiah has no momentum and the state has no center mess momentum and it's a matter of choice and with your think of it as literally being a little spin or just an abstract property that fewer mathematical property that you attach to the electron whatever it is angular momentum it transforms on the rotation when you transformed when you rotate coordinates the state transforms now we've already been through this we spend the most water talking about they have been system summer quickly should go back and read the early part of the year year of previous lectures we talk about us having 2 states upward then but then we said Wait a minute if committee up and then can also be left or right but if we left writer comedian are out there we about the state's describing those things the operators which describe the components of the Spanish and I didn't tell you what we were talking about the angular momentum of a spinning particle are very have spin particle so let's see How we can understand that now that we've talked about angular momentum the thing characterizes angular momentum is 1st of all that there are 3 components of it but just a factor of 3 components of not it must be that those 3 components are literally associated with the ex-wife Kinsey directions affects Ukiah have been connected with something else some other internal dimensions or whatever but wants to have matrices once you discover they use system is being described by matrices 3 of them which have the commutation relations of angular momentum and they are angular momentum 5 actually has no choice about that as long as the 3 matrices you're talking about a literally associated with acts of Hawaii and Izzy direction space so only write down again what the basic commutation relations of angular momentum are L I J unless right just
L X armed but state Elzy Z With L X Els of LX logistical top 5 times Hill White any of other 3 of the other 2 are just a quick permutations OK you may have noticed that these kind of relations are closely related to the commutation relations of the alley matrices so let's write the 3 Polly matrices and check whether this is true of the Republic matrices why should here they are said
Missouri is a global 1 minus 1 0 0 now we've made a choice we made an arbitrary choice we decided to take signals Zito the diagonal matrix of other words we're working in the EU representation of disease component of the spill which I wasn't take That diagonal and of course the Eigen values were the eigenvalues of this matrix beguine was matrix of plus or minus 1 representing opened OK then there was X X woes 1 1 0 0 and finally similar why is eagled the minus ii I 0 0 there ambiguities in these Yukon rotate them and change a change them but there is no nothing Durham reported half which is not normal users in Rex all 3 of her Hermes a real matrix meaning to say it it's entries are real numbers the Commission its symmetric but as this and there's never matrix is her mission and an imaginary and must be enticed so that's this OK let's check the commutation relations let's multiply city Lizzie by Sigma X and that's equal to 1 minus 1 0 0 0 0 1 1 0 this time this 0 this times this is 1 of here 0 here minus 1 0 0 that's not quite signal white but almost certain Hawaii it would be signal wine I think um is a bike time signal wine for minor cuts are like time signal white young now as my but it's not the commutator a commutator is this my stigma X soon that's the county and that has 0 1 1 2 0 if 1 0 0 0 minus 1 and if you don't throw it 0 40 get exactly the same thing you have over here Yukos Arsenal white so we have not quite what we wanted we have instead commutator of singers EU with Sigma X sept is twice more why does this mean these among angular momentum of doesn't mean that we are talking about rotations know it just means we normalize the signal matrices incorrectly amiss here's trick take S and I find it to be said tool for each component for each component take as we have the Polly matrix odd but nobody told us our work for the angular momentum was supposed to be so it's just take it and see what we get them we get so let's put it to India the tool here and not altogether divides by 4 right for it so here till here I've divided the whole thing by 4 so divided but for is one-half so yes have the the Polly matrices satisfied exactly what the supports the satisfied and Z with X is equal top might SYT likewise for the other commutation relations so we say this little system off 2 by 2 matrices acting on toe component vectors upward are actually representing a very primitive and simple angular momentum system our it's a thing which of the think attached to particle the catch the particle will spend and will spend simply described by this system here but now we can enter the question What are the ideal values of let's say the Z component of the angular momentum that's got busy component of the angular momentum component of the angular momentum clues here now that we know what we want 1 have minus I have won a F F 2 2 Norris that day I give values of z component of ideal moment but ahead and might Hey we will working out angular momentum I told you I showed you the dealer 2 kinds of multiplex the weather have been multiplexed 0 over here sorry military integer spin multiplex where the magnetic cornerback and woes 0 1 2 3 miners Foreign Minister minus 3 and so forth With the central point being at 0 being 1 of the possibilities member they also grazing and creates raising Moran operators shift you an integer but that the started 0 2 EDT start with 0 or you start that I had F but with but half minus a half and then 3 halves minus 3 heads depending on the overall angular momentum if it's a half the was only have murders but good and so we
have here we could have guessed all of this by saying what we see that the mathematical possibility of having have spin that means there 2 states once we know 2 states we can say the angular momentum of this 1 must be plus any angular busy component was because they have that were might have look goal and start building the structure up the way we did about a year and a half ago or Europe whatever was Laura was was that Bongo OK so every particle it has been spin maybe 0 if the spinner 0 which is called is 0 particle Biggs both arms been 0 particle the Deuteronomy is a spin 0 particle our guests nuclei can be particles are particles they have no angular momentum in the rest frame on the other hand some look where I do have but some Adams has shared but the important thing is to forward concentrate on they have been particles electrons that new 1 neutrinos quirks forms a prairie rethink of this fundamental summer of storms with the fundamental have have spared OK all of this means that there till kinds of angular momentum what will angular momentum this is truly classical physics to what angular momentum is the angular momentum of the center system it mass frame with the system spending has been the same is true here and was the total angular momentum with total angular momentum as the summer them it's a vector so the vector sum of the 2 kinds of angular momentum and it's true here so I'll write down the equation but do not do very much work Carlo angular momentum is called JT again it's a taste overworked comes from its again a vector and equal to the orbital angular momentum L which is forest P Plus spin class S just tell you that because you'll see that all over the place when you don't want the mechanics particularly in the atomic physics you'll see that total angular momentum was called Jerry but L pour search for particle any questions about spin there a bit too much so we can stay with for a while but I hope we can I heard it the success of the G-8 are usually given usually given something different usually given the knowledge the spinners a half for example and that the orbital angular momentum could be 1 0 at the orbital angular momentum 0 the only angular momentum it has suspended if the orbital angular momentum as 1 we need some rules for what we get if we had or angular momentum 1 to an angular momentum over half questions what you get and I was going to that now the problem of the addition of angular momentum it's a straightforward problem but we won't do it tonight they happy go through it overnight but there are rules mathematical roles a adding angular momentum of our health 2 systems basically you have to systems of the spin system and the orbital system or you could just have to or you can have 2 different orbital angular momentum systems are the system has a torn angular momentum and there are rules quantum mechanical rules for ending Mary her it's not do it now a list of all of us as if had last a fall high Lois drawn that these are back over his head of the 1st experimental background came partly from spectroscopy and partly more portend from our from periodic table so that's what I wanna get I wanna get the periodic table a little bit and show you how spin comes into our Polly who realized don't forget To understand the periodic table electron had have an additional property OK that brings us to the issue of Fermi Meaghan's involves arms which is closely connected to meet Polly exclusion principle at least in the case of Premier Andreas the images motivated In terms of chemistry a little but it was 1 of the motivations 0 the came from spectroscopy but it's it's not as clear obvious these and remember the the angular momentum squares L. squares is equal to L. plus 1 that what I want to be out there and have if we look only look at these levels be energy levels Of the armed of these hydrogen atoms solution which we didn't do but I just told you a fact a toad affected some extra D generous each 1 of these has To well plus 1 state but there were additional deadly generous use of only draw them for you a horizontal axis is hello the vertical axis is energy is the energy levels of the hydrogen atom and that at the bottom the ground state of the hydrogen atom with the lowest energy level all and it's S wave meaning to say it has no angular momentum built orbital angular momentum the solutions of the Schroedinger equation images of fact negative you start some negative value minus 13 . something electron volts but I'm just putting it here arbitrarily of a zero-sum was appear OK so that a 1st now at these illegal 0 there are states with this is a state with no some away fortune and this 1 with 1 node to those 3 nodes and so forth so they're more states here and has drawn schematically completely schematically they're more fact closer and closer together as they get appear nearly equal 0 but that's up to important for us now we go Taylor goes 1 equals 1 is the lowest energy state of illegals want it each Ehrlich equals 1 state has 3 components to oppose 1 of 3 components in the ground state of illegals ones has more energy than the ground state of illegal 0 White because it's kind of momentum so cost us more energy so it could be up a little bit higher in fact factors hold collection of levels here is good you raise up because of the angular momentum and worries that other surprising end that West was it stays
elegant Maastricht elegant sewed it is what it is I'm sober 1st and the 1st L. equals 1 level workers at the same place as the 2nd illegal 0 level steam generator with source 3 states over here and then the pattern continues the next 3 states gender or race and so forth we generally means it has energy the you LA caused or use a different colors you still which is earlier I want to find you solve the equations this is supposed to be the same level here is you find begins the 1st LA caused to state is higher than the 1st illegals 1 state but happens to occur exactly where the 2nd aliquot to status which is exactly with their illegal 0 stadiums to fall but you know what I mean Europe to here how many states and 5 1 2 3 4 0 5 1 2 3 4 or 5 and so forth and that's a pattern that you can count the number of states that you count the number of states are 1 here with this energy level whoops got 1 here 40 0 angular momentum was angular momentum 1 form not even here 16 year evidently not ill squared levels that you look at the way electrons fill the shows an idealized very idealized chemical description of the arm of the periodic table with that but there are you see that 1st of all hydrogen which has 1 electron when you look at helium which has 2 electrons you find out that uh that we consistent that's consistent with an energy which would be what you would get if you put both electrons into the ground state of things are a bit different be costs instead of having scored a helium just as hydrogen but matter helium helium has a nucleus of charged to cubicles charged to so it pulls on electrons the hall structures Reese but that's unimportant the charges bigger that polls electrons in closer but the basic structure is the same that look at helium you find out 1st of all a helium ions helium ion means 1 electrons there to electrons any study the helium ions and the helium ions looks exactly the same the army as a hydrogen with the exception that you have the plug-in twice much charge that thus physiology levels out Reese scales them but saying structure of Ireland looks like exactly what the heart and the ground state of the helium ground state of the helium ions looks as if it's 1 electron in the ground state of the helium armed with function that you put another electron what do you expect while the natural expectation would be if you put another electron in the ground state of that will be put that electron also the lowest energy state why do anything else you wanna keep low possible all of the ground state so you tried at 2nd electronic into the ground state them away from works just fine gives you the right rough description of the helium is not quite exactly right because it's ignoring me interaction between electron just pretending electrons just experiencing let's take that as a working yeah approximation ignore the interaction between electrons put tool electrons into the ground state of helium wave function and it what did you write energy for in what what happens if you want to make a helium ion by putting in extra electron instead of not instead of a B B I am with only 1 electrons is also with no electrons just call army doing nucleus OK 1 electoral with free electrons in what's the against right into the ground state why anything else but no that's not what happened what happened is it goes the 1st excited state Super Tour electrons and helium into the ground state and the 3rd 1 doesn't want to be there for once they get into the next excited state wine it could save energy you could have had a lower energy state in the ground state that electron went into the went into ground-state OK so along from Cali and says I have an idea the idea is that no 2 electrons can ever get into the same state something about electrons is such I won't call a repulsion that's not right the call the repulsion it's an exclusion principle when electrons in the state can't hit another warning somehow the mathematics former wipe away a minute we're ready said we could put 2 electrons until helium ground-state sewers talking nonsense politics was in principle doesn't work only work somehow put a 3rd 1 palace that's not what's going on he said really you can't put 2 electrons into the same state but electron has more properties just its orbital motion he said will make up the new property and no property has to be such that the electron condoms With tool values of true values are in that case you could put electron 1 electron into the ground state and let's call it we are up electron member there actually can now have 2 properties of cold up and down column left the right anything we want our about let's give them the new property which a Paoli eventually realized had something to do with angular momentum and said it put tool electrons and state not be costs EUR out you can't put 2 electrons of state but you could put 2 electrons into the seam bowler Biddle angular momentum state if you give them opposite Spain if 1 of them is put in the upstate and the other 1 was put the downstate then I can save my crazy exclusion principle I could say like crazy exclusion principle of I assumed that the electron has 1 more property in addition to its motion ambition total orbital motion that has things which I will call and now back there was to a previous reasons to believe in angular momentum like that but they're not as transparent from award treading into their loan that was as basic reason and electron has another property but to value property once know those To valued property use often going he invented matrices to describe that to value property and before you know it he had the Cali matrices and understood the whole thing but our what's cultivated let's go through next Adam to we go through a lithium lithium has 3 electrons would naturally expect for lithium could do the same game now nucleus has charged resource much more attractive charged 3 you put 2 electrons into the ground state that work there for a video of the no electron case the 1 electron case the 2 electrons case lithium with him ions it works perfectly well the put electrons into the ground state that is not electrons prefers 1 2 nd you find out that when you put the 3rd electron in its ability to
make it neutral it goes into the 1st excited state because in the 1st excited state meaning the state with angular momentum 1 that works out well works out the energy levels of road of the U. lithium Madam worked out reasonably well but wary about the ion of lithium well filled up the 2 lowest energy states you only put you have restates here you could edit all nuclear another electrons in any 1 of these states with go back a step but Galimi go back a step but forget for a moment the spin and let's talk about the young and the atoms of the was no spin but if it was Powell exclusion principle Tripoli is all idea except for artist would you foreign did you would find tool will you find 1 helium and helium you would find 1 electron things right In the ground state but then keep the energy low you could put the 2nd electron in any 1 of 4 states in any 1 of 4 states if it in 1 would double Romano violate the exclusion principle it would be for a distinct helium atoms 4 distinct helium atoms depending on which they put the next electronic that was not right the helium ground-state state was unique did not have this peculiarity what I always idea was not you don't go to the 2nd stayed here and there are enough for possibilities this just a helium atom with 1 electron and helium Enomoto electrons both from the ground no you Yukon prolific Yukon Poephila you put the 1st electron forgetting Spain where commit 1st electron here I'm getting cutters usual form trying to do is explained why there are 8 different can histrionic table of there the 2nd row for as 8 entries high hope has 8 entries intrigues how we understand entries said with a thought we had the 1st 2 electrons and put them in the ground state and that the that's the I will finish programs state then emitting more states we have if we only occupied the 1st excited state and or a timely possibility different ways make Adams well we can have we can put an electron in any 1 of 4 states in the 1st excited state but if would allow the play with a spin and anyone 8 any 1 of 8 possibilities you can say is that of sex was quoted as team we are also gave restated their solicitor let Johnson a hero for state to the 1st excited rebel C D 0 Europe them 0 three-ring your more things that they say is a new year freedom of Carrefour for 4 of Mario Double would buy was spend it becomes a yes you get fully that says OK that that's how spin was discovered that tells them was discovered and the fact that really was angular momentum that was that could be checked by putting me magnetic field electrons additional degree of freedom was really angular momentum that would mean that electron was charged that the electron would be behaving like a little mad it was really Europe's spinning and it was charged it would behave like a little over like a Mac like electromagnets current course around the globe if that were the case and you put the electron until a magnetic field preferentially it would be it would lie along the magnetic field Our opposite to the magnetic field with 2 different energy levels took the hydrogen atoms with 1 electronic and now you put in the magnetic field the 2 possibilities for the electron spin along the magnetic field an opposite magnetic field would now have different energy levels so the the tool states of the hydrogen atoms which previously had been comply ground-state ground said of the hydrogen atom with electronic electron proven pointing in 1 way or the other way they have exactly the same energy in the 1st approximation and they put the magnetic field and you find the energy levels a Split and this Split exactly as if the the electron little magnate because of its angular momentum so the fact that had angular momentum that to be tested by putting a magnetic field the fact that had to levels that was consistent with 3 with the public schools in principle and fact in our or be assumed fact that you can't pork 2 electrons in the same state that this was all all while guessed it was while guessing make could put 2 electrons in the same state it was wasn't quite so while but it was angular momentum Caestecker B tested experimentally but on the next step which I think was Pauly it was below the 1 person that it wasn't was the person that the idea is named for fairy but the erratic and must interact and my colleague the polio Barak who had the idea that there are 2 kinds of particles particles which satisfy the Paoli exclusion principle will you couldn't put 2 in the same state and particles will you could put 2 in the same state cannot only that it was preferential to put more than 1 state found it was already known the photon photons for our purposes now just particles was already known that classical wave show times was simply a collection of a large number of photons all in the same state called the same quantum state so it was known that the particles but you could put into the same state Einstein forget about him again was named as the Bozo because Einstein forget about and the car was although both that was a race what really there was sit through now he came later and his contribution was different from the discovery of the awards called the statistics of particles their bosons the bows on Australia character was Einstein and Pauly I believe what might have been Einstein Barak in Cali and not sure what the actual history woes and it was not people they made major contributions but was Einstein who gave party both the credit the discovery don't both of their particles talk about software with associated with high but
just as them but to a missing state that's why you can make a laser that lasers electrons laser creates lots of photons old strained quantum state power you would have a pretty you wild if you could get electrons in the same state do the same kind of things with them but you can never remarkable microscope that you could you have electron microscopes but this would be a rather superior electron microscope is so that all of these is not related to rotate this shared course there's a star-crossed they are particle a particle moving in free space is moving along line can have an angular momentum on gross P. relative to some origin get have out the angular momentum is relative to an origin as because Aura's relative toward origin definition of Aura's relative toward fixing fixing in origin UNITA and the whole idea of rotation is about some origin the whole idea of rotation picked social emphasize at the beginning when Elliott talking about angular momentum of talking a rotation of picking a special point the study of about 90 case cases and it's natural to pick the nucleus to be appointed to of a point of symmetry Our for particle moving in free space every point support circuitry a rotational symmetry so not it becomes a less interesting new thing but for the young the Adams is a natural center talk about rotations about he should emphasize that but is a liar if she Rosa 8 elements that Rose they said on the way to at some point the whole picture bricks Barry was McKinney said I'm pretty quickly in fact the picture of shells and filling shelves is more than a little bit naive it completely overlooks the interactions between electrons them by the time we have a fairly large Adams is basically as many electrons in the in the quarter as there are protons at center so to ignore the interaction of the outer electrons would be in electrons makes no sense at all over the whole thing breaks down pretty badly and then and then this is zillion rules replied I don't not bad we did they didn't get bigger practically out of all the reason is because more electron what happens it is very simple more protons you have the nucleus the larger the charge that can stop poorly in the core of electrons the either shells of electrons and with leftover if you have with say 1 valence electrons is 1 valence electrons moving in what kind of field the field that it's moving it is it is protons in in mind this 1 electronics and salt that 1 extra valence electrons basically thinks it's a bit of thinks it hydrogen and it's no bigger than hydrogen really essentially on the same size as hydrogen sold for my better think that circulars right Adams not get bigger appreciably big it's very slowly but they don't get appreciably bigger issue lecture as you a protons and electrons of time I there is a there is that they say as far apart from each other is because a lack of hygiene or reverse a report from forward from power rift whose approachable that exception to the rule is most Adams however were poultry at its what OK warily gaps we talk about bosons and fermions this viewership said Is there a few clues that a separate that shit now not know it's part of basic special relativity together with poor mechanics quantum mechanics without special relativity it's a postulated but the postulate can be phrased the more elegant mathematical way that's Silverado Frazer postulant more elegant when want to introduce special relativity you combine it with core mechanics it's not a postulated it's a consequence of special relativity which was the racks discovery OK said a talking I want particle with talk about 2 particles well let's back to classical statistical mechanics for a minute is classical statistical mechanics on classical mechanics of of a system of particles of hope articles of Oceania kind of particle the question is beat treat them as as identical particles in the following sends cannot in the number of configuration for example while we put particles in the box the boxes could be a little boxes in phase space circuit as big boxes but 4 boxes wikibook particles but say we have to particles and that both the same kind of particle so here's is a configuration weaker but don't 2 them both into the same box that's clearly a unique there a unique configuration or I could put 1 particle in 1 box and other particles other box is that 1 configuration or is it to configuration are these particles had means Harry in selling Harry and Sally there have means the Harry years Sally hero you could put Harry selling here in this seemed to be 2 different configurations 5 encounter configurations in calculating and could be in store what this could be dead this sounds like it's a difference now whether it is important difference on was a separate issue but do you can't these separate configuration you could always imagined that these particles which are identical to you have a little trace little bit tea them since then leaned on but it's such a week the pain classical not quantized pain classical painted itself sold terribly faint that no experimented you ever could do could be To date I mean imprinted on the particle but you have to say they would different particles Harry here Sally there would be different this year and a half and Harry their arms all are you could say no I'm going to do statistical mechanics as if these for the same configuration not think of the particles labeled not fight to get the same marriage which doesn't make any difference classical statistical mechanics but conceptual different putting a particle here a particle here Is it or is not the same switching them if they're the same kind of particle so that's a classical version of the notion of identical particles in a classical statistical mechanics we usually assume that these are the same configurations Harry Sally Sally Harry configurations and that's the way we do our counting how many configurations are there for a large number of particles well typically you
get some in factorial different ways of re labeling the particles and that goes into statistical mechanics of you taken statistical mechanics course you know that partition functions there these won over and factorial as simply keeping track of the fact that a if you interchange to particles in the same configuration that's a researcher made in quantum mechanics you don't have the luxury of doing 1 or the other it is very definitely that particles identical in the sense that they do not carry labels to electrons 1 electron here of electron here is mathematically identical and you could do it otherwise our exchanging your terms similar to a photon is now having a full-time here electron here is not the same as having electron here default on it's clear that that should be out there very different properties in Ucon Contos 2 configurations apart but it did changing in swapping around which electrons which that's a nun transformation on the state of the system and also was true of photon so let's think about that and what it tells us about the quantum state on a pair of identical particles of stick a pair of identical particles as could be cool electrons and for the moment forget love spin or anything else we just have to elect firms in the in some states but she gets early described to particles 1 particle is described by a wave functions sigh of X where X is the position of electron or not electrons manufacture puts the go back that was the sigh of extras meaning and see overlap body in a product of an electron X with the state of the system side of sigh is a state vector of electrons minutes projection or in a product on was state localized Texas quotes I suppose electrons heading characterized the states of 2 electrons will you characterize them by 2 positions position 1 position of the other let's column X 1 next to that does not stand for different directions space since the 2 electron excellent next to you and the wave function is a function of 2 coordinates instead of 1 of X 1 and next to that that's the way function of 2 particle system a function of and if you want the probability the probabilities of probability defying particle won position x 1 particle position next song we have functions of 2 variables when we have to particles now let's think of the operation of swapping the 2 particles taking particle 1 and replacing it by particle to particle too but particle 1 wants transformation that's a transformation that generally will change sigh of X 1 next to change it From sigh of X 1 Mexico toward the sigh of text to Annex 1 which is not the same thing in general a function of 2 variables does not have to be symmetric it does not have to be such that if you interchange to arguments you get the same year the same function back so in general a transformation of transformation which swaps too particles let's call at transformation with given me it's an operator operates on a wave function and gives a new wave function so let's let's name it was called swap operation Court S. & Hill West does yes state With particle 1 position x 1 and particle to add position next to I know it's particle 1 particle 1 comes 1st article to come 2nd What does the swap operation do when it acts on particle at position 1 2nd particle position to just swaps it gives you particles to warn started particle 1 at position next tool and particle to end position x 1 2 swaps and if the particles had little means attached to them this would mean something have Harry position 1 and Sally a position to Liu would have Sally a position to and Harriet must right but you know what I mean to change of just over banks and those switch them so this is a possible operation that you could do what on the state of 2 particles yeah that interest in fact take the operation as square was a square as swaps in the well did the swamps of swaps again what happens if his swap likes it that same thing obviously back and playing even a half the particles are little means attached interchange Harry sodium interchange Harry Sally again you get back the original meaning so in all but a square as squared must be 1 as an operator Esquerre must be 1 it's a unitary operator why should be a unitary operator because old transformations on the space of states a unitary so his who will acts as his unitary that means it doesn't change probabilities doesn't change on our inner products and squares equaled born where the Eigen values of matrix on operator who squares warned you 1 0 minus 1 the only possibility as squared is equal to 1 means that the Eigen values of plus 1 or minus 1 OK let's what make a new principal no principle is that electrons or that identical particles a Kobe photons could be electrons whatever that for a specific kind of particle the wave function when you when you interchange to particles always comms back the same way functions that is equal to 1 or comes back who would negative if as equal to minus 1 see what that means but thank the case plus 1 I think I'm beginning to fade I think it's time to quit you're probably also began fade will will go through me and borrows arms next time but the basic idea is classifying the wave functions of particles in terms of what happened when you switch the particles to basic quantum mechanical process are quite basic quantum mechanical operations the sway the swapping of particles and the notion of identical particles is basically telling you that something simple happens when you swap particles by the way the function of the proposed that goal will do it for more please visit
us at stanford . ETA
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Metadaten

Formale Metadaten

Titel Advanced Quantum Mechanics | Lecture 4
Serientitel Lecture Collection | Advanced Quantum Mechanics
Teil 4
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 10.5446/15098
Herausgeber Stanford University
Erscheinungsjahr 2013
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Physik
Abstract (October 14, 2013) Building on the previous discussion of atomic energy levels, Leonard Susskind demonstrates the origin of the concept of electron spin and the exclusion principle.

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