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Lecture 11. Particle on a Sphere, Angular Momentum


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Title Lecture 11. Particle on a Sphere, Angular Momentum
Alternative Title Lecture 11. Quantum Principles: Particle on a Sphere, Angular Momentum
Title of Series Chemistry 131A: Quantum Principles
Part Number 11
Number of Parts 28
Author Shaka, Athan J.
License CC Attribution - ShareAlike 4.0 International:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor and the work or content is shared also in adapted form only under the conditions of this license.
DOI 10.5446/18889
Publisher University of California Irvine (UCI)
Release Date 2014
Language English

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Subject Area Chemistry
Abstract UCI Chem 131A Quantum Principles (Winter 2014) Instructor: A.J. Shaka, Ph.D Description: This course provides an introduction to quantum mechanics and principles of quantum chemistry with applications to nuclear motions and the electronic structure of the hydrogen atom. It also examines the Schrödinger equation and study how it describes the behavior of very light particles, the quantum description of rotating and vibrating molecules is compared to the classical description, and the quantum description of the electronic structure of atoms is studied. Index of Topics: 0:01:47 The Solution in Phi 0:03:10 The Solution in Theta 0:24:11 Energy Quantization 0:30:41 The Spherical Harmonics 0:33:26 Transitions 0:37:20 Spin 0:42:21 Stern-Gerlach Experiment 0:48:11 Electron Spin
welcome back to camp 131 a today were gonna pick up where we left off last time we're going to continue our exposition of a particle on a sphere which is going to be our entree into the theory of angular momentum that later on we're going to use when we talk about Real and today than particle on a sphere and angular remember that we had our separated equations we had by dividing by the product of state of status and Fireside we ended up with a sum of several terms 1 of the terms only has 5 in the function of fighting and the other has only failed and therefore by the same arguments that we used in the two-dimensional particle in a box we can seperate the but solutions and deal with them separately and that's of course a very important simplification because that lets us deal with just a function of 1 variable and regular derivative so let's look at the 1st term than the 1st term as 1 over 5 days of fighting times a 2nd derivative and that term is exactly like the solution we had for the particle on rings so this shows the wisdom of doing the particle honoring 1st because now we simply recognized that we've basically got the same situation all over again and we don't actually have to solve it but we anticipate that we're going to have a quantise Asian conditions because we know that we have some sort of a balanced system and we saw before that we had a quantum number and that I called the magnetic quantum number will seek Wyman and now I'm going to call this new number ends of L To keep track of the fact that M may depend on something else and of course since I already know the solution I can seem to be rather Prashant about what notation to pick but anyway the differential equation and the Eigen value equation is that the 2nd derivative of firefighters equal to some constant minus an some Bell squared times 5 and here and Sevele is a positive or negative images and it could even be 0 just like it was for the particle honoring so our solution then is going to be easy to the I and fly and I won't bother normalizing the United function because will normalize the whole I can function over the entire sphere at the end but we know that this makes sense and this is basically what we saw for the particle and honoring the other part than if this 1st part sums to minus animals squared then the other part apartments some two-plus animals squared where am Allison images so we have this rather in In the intimidating I guess the differential equation and fatal was signed faded and the derivative of the function applause epsilon signed square there is equal to Square and at this point in the text in my opinion kind of Dodgers everything and just says well mathematicians seem to know how to solve this kind of equations and while that's true I'd like to do a little bit of guesswork as to how we might try to solve this kind of equations and see if we can get a little bit more insight into what the solutions are looking like by trying to do it and 1 way to try to do it is to say well maybe it's sort of like a particle honoring again and to try similar kinds of things and that would make sense because we have trigonometric functions in this differential equations and so it probably it is true that the solution is also going to have trigonometric functions and because we have this fear this round then we expect that the wave function will have some period is to be with respect to the fate of like it does with respect to fight and that's another argument as to why we can't do that so rather than simply quoting all the results and giving them on a table In leaving none the wiser as they say I'd like to go through just a little bit of of the thinking behind it and so let's go back then to the original equation which had both data fight and we have an idea what we're going to have for of the 5 part and we want to see what we're going to have for the thing part to simplify the notation I'm going to kind of go backwards and not write it as a explicit product anymore but I'm going to adopted notation that's actually used for the solution which is why our of Fader so and that makes the equations easier to fit on the slides as well therefore what we've got now is minus stage bars squared over to Emma square 1 oversize square the 2nd derivative of while and with respect to 5 plus 1 over signed DD theta signed data the set of the 1st derivative of wireless with respect to the data that should equal some energy as yet to be determined which we expect is going to depend on Alan M. and we'll see what they are in the 2nd and we know we don't again times wireless and the key is we have to get on energy that is independent of the variables and fight if the energy is some variable function that we haven't solved the item value equation we just have to get some constant some number and then look at it and see if we've got it right I m was 0 0 I was 1 of the solutions for the particle honoring just flat probability all around and so I could try just using a constant for a while and and have a look at that and see if that might actually solve this particular differential equation and if we assume but the and part has to do With the sea component of the angular momentum that's true because the angular momentum follows they are crossed the rule and so we we know that if it's going around in a ring in
the ex-wife plane which is what we had for the variable fight that is really busy component of angular momentum that it makes sense that the rest of the staff has to do with the X and a Y components of the angular momentum those must be in there and the question is could those b 0 as well if we pick a constant and there's no curvature anywhere than the energy is going to be 0 so we tried and we call y 0 0 because whatever powers in L. we have potentially setting them constant which is the same as raising them to the 0 power and and we know is 0 that was the constant solutions for the particle honoring so that we just put y 0 0 fate of fire is just equal the some constant we don't know what the customers is because we haven't normalized the way functions I think if we normalize the wave function as to get the constant we just put on y 0 0 square modulus and then we have to integrate over 5 from 0 to 2 pie at around the ring and then we have to integrate but we have to remember 1 we integrate faded to remember to look back to the volume element for spherical polar coordinates and include signed data so we integrate signed dated the from 0 to pie and if we do that we find we get for pi times y 0 0 squared is equal to 1 and neglecting the phase which we usually choose to be just real y 0 0 verify should just be the constant won over the square root 0 4 I and as I read as I remarked don't forget the the sign when you do the Senegal's it's easy to do so if you are actually looking at the volume element and just deciding to integrate to normalize the function if you do that you get another factor of Part and everything gets all crossed In this case for a constant the angular energy is 0 because if we put a constant and we take the derivative with respect to fight we get 0 and if we take the derivative again we had 0 and the same with data we get 0 so we just going 0 for everything for an atom in this state which will see later is called S which has to do with some historical observations in atomic spectroscopy that I'll explain the energy the energy the Indiana must be in the radio part because remember when we converted the kinetic energy for the particle on a sphere we threw away the radio parts we just said Oh well where we're dealing with a particle on a sphere here and we're going to throw away the radio part because our fixed and of course that the same reason why that's an artificial thing to do with the particle on honoring although it simplifies the problem is an artificial thing to do here so we don't worry too much about it and that the energy is 0 and that means that the uncertainty principle would would be violated if we I believe that because we couldn't have any fluctuations in Connecticut and the square the kinetic energy them or we would have some sort of energy because that's a positive number now is that we have no idea other than this argument that a constant seems to work what might work 4 the other solutions and data but what we can do is we can just try the same thing that we tried with 5 we can try e to the I faded what happens if we try that and I will let you do that as an interesting problem try putting in indeed to the isolated and see what comes out and usually what you can find out is that you get something close to what you want but it's not quite what you want it has some extra parts and then usually by inspecting those you can get an idea of what to throw away that's in fact what I did on the side of on a piece of paper and then I decided I should try just co-signed data let's stick with Pfizer constant let's try co-signed data for the faded dependence that's a function that has a node and therefore we would expect that that would be sort of giving us some angular momentum in the X and Y the components because now we have some curvature with respect to along that direction if we put that in the end that's going to actually be called Y 1 0 1 we finally get done with figuring out what the proper notations for these functions for now we just put it in and we set up our Eigen value equation same thing 2nd derivative with respect to 5 1 over the sign square data and so forth and the funny derivative with respect to the data that work gonna work through now we just put through "quotation mark data since there's no flight attendants at all but that partial derivative managers we just still control that part out and we have minor bars square over to squared 1 over signed the deal by the Fed signed data d co-signed to the and that should be equal to the ecology 1 0 co-signed data because we have to replicate the function that we put in but the derivatives but with respect to that of the co-signed with respect to the is minus the sign if we don in the next line but take the derivative of minus signs square data that's as you squared and the rule is minus 2 you the EU DX and so therefore I get minus to sign data co-signed data and now magically the 1 oversight data that was out front on this tour cancels the sign data that I got by taking the derivative of Science squared data and I just end up with co-signed data and the minus sign cancels the minus sign on the beach bars squared which is good because I want positive and I end up with 2 wage bars squared over to mr square co-signed the which is to age password over 2
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
evil ones is as big as it can possibly be the top value of and there's still a little extra so there's still a little wobble left and that is exactly in accordance with the uncertainty principle these functions while Emma well known as our textbook says and the called the spherical harmonics and you can look them up and you can read all about them they come in from all kinds of differential equations when you take the divergent said the gradient of the scalar field you end up with these kinds of functions but as far as we're concerned just like signing co-signed or ETI 1 of the 6 Our fundamental things for linear motion there's or waves moving on a line these functions y el land are the same kind of fundamental bedrock starting point for anything that has to do with waves on a sphere as opposed to waves just moving in a one-dimensional and their Eigen functions of the operators L had squared the Eigen value is well times L. plus 1 times H. Baer squared and simultaneous Eigen functions of LC and of lying value there is is page book there are tables of these functions but please please please do not ever tried to memorize all the spherical harmonics if we need them on an exam or need them to do a problem look them up don't try to memorize what they are understand where they came from but don't try to commit to memory all the constants of whether it's minus plus a plus-minus or what the exact trigonometric dependencies if you understand where they came from how the normalized and what they mean and you can use them to actually do something that's plenty you don't need to commit them to memory and again if you take a more advanced courses there are in fact meet waste to generate them so you can generate but these functions by taking derivatives of things and you just start with 1 and then you turn a handle and you get the next 1 and then you turn a handle and you get the next 1 and so forth and of course knowing the generating function as it's called would obviate the need to memorize anything except the generating function and for that you need to remember which generating function goes with which set of polynomials because not surprisingly there generating functions for all kinds of polynomials and that's of course how they're made as we're gonna see the fact that this angular momentum is quantized comes in steps it is very very important when it comes to understanding atomic spectroscopy and atomic spectroscopy is very important because the observation that Adams gave discrete energy lines of light and not just a continuum was 1 of the crucial things that that led physicists to believe that they didn't understand what was going on without a they had no idea they had to the classical theory the Adams spiral away it could have any old energy and then when they got these lines on film they were discrete lines and they were always the same and they had to do with what kind of element you have and that was ever so useful because that meant that you could find out what kind of element you have and if you get a telescope and you look out as far as you can look what you find out is that the whole visible universe seems to be made out of stuff on the periodic table that may not see it seems surprising now but that didn't have to be it could be that if you look far enough away that there is something that you don't understand there some chemistry occurring in some blast from some star left over that you don't understand but when we look out we see molecules that we see on earth we see hydrogen everywhere in the universe we see all these beautiful lines and red glows from galaxies and it all makes perfect sense because we can do the same experiments in the laboratory and see these lines and so it seems like there's nothing but mysterious about the stuff we can see that stuff we can't see so-called dark matter could be some different different story altogether because if you have something that you really can't see at all or not very well then and it's very hard to figure out what it is because it's more less invisible to the photon has 1 unit of angular momentum and because it has an intrinsic twist and that's really important because when the photon comes into the atom the photon is annihilated the photons here the photon is gone but the angular momentum we believe to be conserved just like the energy and therefore it must be that the twist that was in the photon is converted into a twist in the wave function for the atom and vise versa so when the atom emits a photon a twist has to happen and the photon comes out and that that requirement that angular momentum be conserved along with energy really simplifies the atomic spectra otherwise we would have Chileans of lines and it would be virtually impossible for us to figure out what was once but because we only have a few and most of them are missing it was possible for Redbird to see the pattern in the energy levels I therefore if we have an atomic transition is we're going to see when we talk about atomic spectroscopy in more detail there is a selection rule the selection rules as Delta Air little elves from the spherical harmonics is equal to plaster minus 1 In other words the photon has to remove for absorb twist and the atom has to compensate accordingly there is another player in
the game called spin and it doesn't surprise us that if we got charged particles like electrons and it's in something that we understand to be honoring their on sphere is moving around circulating that it looks like a current lieutenant has a magnetic field but the experiments that were done more detailed experiments In cases where the the Adam should not have electrons moving around because this should basically be in this constant state this and equals 0 kind of state that doesn't have any twist and that can have any angular momentum they still saw but there was some magnetic phenomena and the pivotal experiment much like the "quotation mark photoelectric effect was the quantise edition of light and the Davison Girma experiments showed that electrons could behave like waves the stern girl like experiments showed that electrons themselves we have an intrinsic magnetic field In addition to charge and mass electrons also behave on their own like the little bar magnet and because we know that but articles on a sphere and Adams will see are quantized according to these equations with units of age bar that would make sense that the electron would somehow be quantized nowhere with this and magnetic field becoming from all fleet if we blow up the electron we have some kind of a planetary models Florida a uniform sphere of charred then if the electron were spending then this uniform sphere of charges for just spinning and maybe it couldn't stop I would create a magnetic field because there would be a bunch of current loops and unfortunately that kind of really concrete picture of the electron actually spending but I can't possibly be true and so rather than them saying the electron is spinning the Dodgers to say the electron has spin rather than thinking of it as a mechanical thing actually occurring it's now just a property like charge it's something we give it a name and we it's there and we can't quite explain it in terms of a planetary model of a big blowup of the electrons but as I mentioned before the electron appears to have no size so it would be difficult to figure out what size to assign for this planetary model anyway Stern and girl I used a beam of silver Adams to do their experiment and the reason why they did that Is there silver Adams have a close it basically closed shell electron and closure of electrons when they're all filled up the angular momentum is 0 that's 1 reason why "quotation mark shelves are especially stable and then there's 1 electron and invests orbital and that orbital has no orbital angular momentum and therefore and the spin angular momentum if there is any for all the close shell is also 0 that can be worked out and therefore we've got 1 electron In the outer shell and we should just be able to use a silver atoms than to figure out what the value of the anger momentum this 1 electron hats that's why he picks over the other reasons why he picks over is that you can heat up silver in an oven in a vacuum and you can get a beam of silver atoms you can make silver vapor you have to heat it up pretty hot but you don't do a lot of atoms and then you can send them through a detector which I'll talk about the 2nd and you can get silver stripes just like making a mirror and you can see whether you get the classical prediction or whether you get something else now how do you actually interrogate on something as tiny as a matter of what the magnetic the dipole moment because remember this is like a little Barmak how we gonna do an experiment that gets us to to get this thing to move well if we stick up but the these magnets your uniform magnetic field nothing happens they tend to reorient but nothing happens To them and so Stern anger used a magnetic field gradient not I'm not a uniform magnetic field they had a beam of silver atoms and then up and down and they had a magnetic field gradient it's not easy to think of what the magnetic field gradient does to a little bar magnet because magnetism is a little bit more mysterious than just electrostatics but is basically the same as if I have an electric dipole In an electric field and it turns out that a magnetic field gradient is going to put forces on the little bar magnet intend to accelerated and deflected and which way it deflects can tell us whether the North Pole was up or whether the North Pole was down when it went through as I said it's much easier to see this in the case of an hour an electric dipole an electric field and so I've drawn by analogy the situation where we have a capacitor let's say and it has uniform positive charge on 1 plate and uniform negative charge on another play and so we have the electric field lines going through at the edge they have to bend but let's say were in the middle and they're going through nice and straight we have a uniform electric field in space if we have a dipole which is always negative charge hooked to a positive charge and the dipole it's like there's horizontal in this electric field ill-feeling toward because the positive charge will tend to get pulled toward the negative plate and the negative charge again churned toward the positive plate and therefore it will tend to align but it's not going to move anywhere because the total force on the things from the positive charge and the negative charge and they're hooked
together otherwise it would be a dipole is 0 so it will tend to twist but it won't tend to go anywhere if it's going through it'll twist and then-state going through but now suppose I make a positive charge more concentrated on the town by shrinking the plate so that the felines have ticked converged together then I get this I'm picture shown on slide 302 where now because there's more positive charge even if it's aligned let's say it's aligned with the negative on top the positive on bottom but there's more positive charge here nearer to the negative charge then there is negative charge on this more spread out plate To the positive charge and therefore these 2 will be pulled up all along the field lines where the gradient is getting closer together and that will put a force than on this thing and if it were moving through it would tend to deflect great now let's do the same thing with the magnetic case ballistics silver Let's make a being out of another individual silver atoms in a vacuum of course no error will just get silver oxide in the experiment will be off put through some choppers why because we want to let only silver atoms with a certain velocity go through put him through a small pipes align him up so that we know that they're all going this way and they're all going at a certain speed giver take and then put them through a magnetic field gradient and the silvers and that has the single electron In the file this 1 configuration don't talk about that later when we talk about the electron configurations of events in more detail now if the intrinsic bar magnet of the electron were a classical thing you could just be pointing anywhere I mean why not then we would expect to get justice Brady of possibilities is depending how it was pointed in the magnetic field it would get deflected upper down to exert maximum amount and then anywhere in between but in fact when they did the experiment which have sketched here of landslide 304 the classical prediction would be a smear and when they did the experiment carefully and when I say when they did it carefully I believe historically when they 1st that they may not have had the beam narrow enough and going enough at the same speed and of course if you have things going through at different speeds so there and their 4 different times and you don't have a maligned very much that it's sorta like having a camera or even deliberately blurred the image then you can't read anything just looks like it's all gravy can't read the news print and I think the 1st time that initially they got what seemed to be something that agreed with the classical result with them when they did the experiment more carefully they got to spot but they get to spot that means that m could be some positive things for some negative things but the theory of angular momentum says that the difference between the levels should be integer and the only way he can ever to levels and 3 but I have 1 0 minus 1 only have 2 levels would mean that it would have to be one-half which was really strange because that's hard that's hard to see how how how the intrinsic angular momentum could be 1 half but nevertheless that's exactly but was seen we get 2 spots and the conclusion then is that the electron magnetic moment is quantized it corresponds to some kind of angular momentum that's intrinsic to the particle we don't have approved detailed picture for but we know it's there and the 2 allowed magnetic states which we tend to call up and down the largest M sub rather than members of Bell M sub as for spin plus a minus 1 half and it follows then that the spin angular momentum which is just called s like is just 1 one-half H that's the but allowed value of the spin angular momentum and there is no other solution this is just an intrinsic property of the electron itself has nothing to do with anything except that I feel intrinsic property of the electron and it's very interesting that this was puzzling as to how this could could be like this and next time I'll talk a little bit about spin and how important it has been whether particles have a half integer spin or an integer spin on because other particles like protons and endurance and other things when they looked carefully and there were Nobel Prizes in those fields as well also had spin may also had little magnetic moments and they were a half or sometimes they were 1 the electrons always ahead and they separated into 2 groups the halves and then not have the images and they have completely different properties in many kinds of experiments in physics and chemistry the next time will pick it up from there and continue on a exposition


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