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Lecture 11. Particle on a Sphere, Angular Momentum

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I co-signed data and that's equal to the 1 0 co-signed the and therefore I figured out what the energy of this state is that's co-signed data and nothing in flight it's a nite in function and the Eigen energy is to H. Baer squared over 2 times the moment of inertia capital I I should be careful to say capitalized so that you don't think it's the square root of minus 1 good am but that's just again 1 solution we're sort of in this situation we were with the harmonic oscillator that we could get the ground state and by arguing Well it should be something related to an exponential we try Yedidia acts no good and we do a Gallician and bingo works here we tried co-signed data and at work and what we can do that it is the try signed data because co-signed failed on Sunday the closely related both have similar curvature and what let's look at this them and try signed but 1st let's normalize this way function co-signed data that's kind of a good exercise because we have to do those integral and if we do the integral so we have to integrate over for overdue fines "quotation mark squared because remember we have to take the wave function and square before we integrate and we do the integral with respect to find we just to fight because there's no fly the function of fighting there and then the sign data for the volume element is on front so now we have an integral to do the integral from 0 Part of squared their signed data the theater and we can look up that integral or we can work it out by the substitution you is equal to co-sign data and if we do that we find that that integral is equal to two-thirds and so we have to pry from the 5 and two-thirds from the state apart that should equal to 1 and therefore the constant out front should be the square root of 3 upon for power that's our final solution for another energy I can function that we haven't seen and that we call it y 1 0 because Ellis 1 for this what if we try sign data what it's it's kind of a good thing to just try it and see what happens and here's what happens if we try signed data we then have 4 the term the chain term the derivative of signed data with respect to think that's co-signed data so that we have the derivative with respect to the favor of signed times co-signed we do that's a product so that's the derivative of the 1st times a 2nd which is co-signed square data plus the 2nd times a derivative of the 1st which is minus signs squared data and we end up with these 2 terms and now there's no consolation we have minors age bias squared over to unmask squared 1 over signed data co-signed squared their the minus sign squared data and that should be equal to the some constant kind signed data that cannot possibly be true because we have this coast squared and signed squares and that they are canceling out and so that cannot possibly be true for a constant the 1 thing this very therefore we have to think Well how can we sort of make up for this and the only way to make up for this it is to have some other part come in From the derivative with respect to fly but that means that we have to put in something for the flight path that gives a non-zero derivatives while we know what we're going to put in for the fight part we're gonna put in the 2 the I and 5 in the easiest 1 to try his hand is equal to 1 sense M is equal to 0 as this 1 and this 1 didn't work so we put in as our 2nd attempt the 2 the I fly time signed faded and we put that in now this is the 2nd term gives exactly the same thing as before except there's another term the I-5 riding along so I don't have to do all those derivatives over I can just write down 1 oversight data Co squared data mining sites where data all kinds the I find them there some constants of from now however for the flight path parred the 2nd derivative I get something because I have a fine and that actually comes to the rescue so recall that part is the same constant won over sign squared data times the 2nd derivative with respect to fight "quotation mark David E. to the I 5 the 2nd derivative of that is I squared times need the I-5 because I just bring down the constant whether it's a real number on imaginary number it does not matter In the slightest I bring down the same constant multiply and I follow the rules I squares equal minus 1 great perfect that gets rid of the minus sign on the exporter's squares so that's out of my hair and what I end up with then because of the 1 over signed squared data on front is I end up with age bias square over to and Morris work times won over signed the times EDI 5 and now there's a trick on the train and often times in the Zeitoun value equations there is a trip and until you see the trick you don't think of it because I've got 1 1 is a simple thing but 1 could be written in many many different ways and the trick is to figure out how to write 1 in a way that's useful To add to the other terms since the other term has cos squared data and signs with data and I'm not old the Coast squared data plus sign square Theater is equal to 1 that's what I'm going to use so in the last line on this flight to 93 I've written H. Baer squared over to an hour square times Coast squared plus sign squared the divided by Sunday times to the I fly if we had the
2 terms of we had cos square data minus sign squared their signed data mn Co square data plus signs within over Sunday and so we end up the coast square goes away and the sign squared and and therefore we end up with 2 sides square divided by signed there and then they know we end up with the same function but we had mainly to H bar square over to an hour square signed data ETA the I-5 is equal to the energy time signed data times To the lifeline well that's perfect because that means that the energy has to be that constant to which bar square to mr square and interestingly enough that's the same energy as we got when we used to co-sign data so it seems like at 1st blush the power invader has something to do where the Eigen value for the energy but we get and that's in fact going to be true we have the exact same energy and so that means that the system has to be generous and recall the generosity is is often related to symmetry when we had the particle in a box we had a generous when the 2 dimensions have the same size and in this case we could imagine the all were doing is swapping around whether we have the disease component of the angular momentum or the X or Y component of the angular momentum and they would have the same energy but they would just be rotating about a different axes classically and they would be rotating at the same rate so that seems OK and that's not troublesome another equally valid choice which won't surprise you if you compare keyed to the I-5 is that you can pick me to the minors I fight again time signed and that also has the same manage and will call that 1 the 1 minus 1 so we have he won 1 the 1 0 0 and the 1 minus 1 and there are no other ones if we try signed data or Coast with each of the 2 I fight we end up with the Mets so therefore those are the 3 we know it has to be images so those are the 3 and we start to see a pattern for the bottom 1 where the data parts a constant we end up with 1 function in FY which is also a constant and is equal to for the next 1 where we have some trade functions and data that race to the 1st power signed dated and co-signed data then we have 3 values of fight we have 1 0 and minus 1 and so it looks like the values of em are bounded by the value of and that would make sense because Alaskan that turn out to be the square of the total angular momentum and and has to do which is disease component and busy
component of angular momentum whatever it is you can exceed the total of all and so would have to be bounded by the end of course they come in units of H bar each of them so well elsewhere comes in years of age part squared Elzy squared comes in units of 8 baht square and Elsie Alexa now why come in units of just age by the I'm so the solutions have to times age bars square over to wine then the 2nd powers and that we coast squared and co-signed signed sign squared not surprisingly there are 5 values of them that go with that too 1 0 minus 1 and minus 2 and those are the 4 or 5 and so we see a pattern 1 3 5 and that pattern continues and in general there are always 0 plus values of but a possible to choose where L is the highest power and in the trig functions that you pick for the now without some of the more powerful methods to analyze the equation it's going to take us forever to play around and guests different kinds of functions of
invader that happen to work and could be an an extremely frustrating thing like doing a crossword puzzle rather than doing mathematics where you just knock it out but for for us at this point we are going to go into the theory of differential equations to such a depth because it's pull too far afield and that would be a proper course to take in math but I think you can see why such a course is incredibly valuable because if you come up against equations like this on your forced a sort of tight you can't hack your way through the equations which can be good if you only want a few solutions but it's very slow or you can be certain become the master of how to figure out these equations and move a lot more quickly the cost it takes time to learn how to type and it takes time to learn how to approach these kinds of problems in this effort more sophisticated way so actually can get all the solutions and not have to people spend the time and time writing them down therefore we can have generalize this In the light of the pattern and we can say Look the energy E L M is equal to L I can't help plus 1 Times Beach bars square Over 2 capital the moment of inertia and classically these energy of a rotating body is just L. squared over to Iowa Ellis again the square elsewhere is the square the angular momentum and will therefore what we can associate then is that the square the angular momentum in the case of a quantum objects particle on a sphere is now quantized In units of age bar and that's perfect because that's exactly what we expected the funny thing about it is that rather than just tell squared times age bias squared little elsewhere we get L times plus 1 we couldn't any differences fellas 0 between elsewhere in time selfless 1 but when always 1 when we have some angular momentum there is a difference and there's this extra plus 1 and as we shall see that actually has something to do With the uncertainty principle it just came in out of the equation here but it also is very closely related to the fact that there has to be some wobble that week we could actually have all the angular momentum be along z if we could all be along the sea and there were nothing left over then there wouldn't be any uncertainty in any of the 3 z would be known and Alex why would also be 0 because the square would have to add up and to make sure that doesn't happen we have an extra plus 1 becomes inconveniently another unit of age bar that
Kryosphäre
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Derivatisierung
Herzfrequenzvariabilität
Kryosphäre
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Nanopartikel
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Löschwirkung
Magnetisierbarkeit
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Systemische Therapie <Pharmakologie>
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Periodate
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Atom
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Kryosphäre
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Gezeitenstrom
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Systemische Therapie <Pharmakologie>
Gesundheitsstörung
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Altern
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Bukett <Wein>
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Setzen <Verfahrenstechnik>
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Lösung
Lösung
Single electron transfer
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Atom
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Fülle <Speise>
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Altern
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f-Element
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Wasserwelle
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Lösung
Kryosphäre
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Tiermodell
Phasengleichgewicht
Phosphoreszenz
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Alben
CHARGE-Assoziation
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