The Theoretical Minimum | Lecture 9

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The Theoretical Minimum | Lecture 9
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(March 12, 2012) Leonard Susskind diverges from looking at the theory behind quantum mechanics and shifts the focus toward looking at more tangible examples.
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Lance Steiger University a W
pre-World formal entitlement to the subject of tribute it is absolutely the heart of quantum mechanics but of course Yao I mean I would stay up till now we have really studied the foundations of quantum mechanics and very very simple systems and the foundations have been laid out exposed you know what they are really the only thing left to do now is to study some examples yup in no classical headaches started with a very small number of empirical observation like everything else is a mathematics were more what's the equivalent in case where the basic atomic empirical observation means darkly balloon Little said you don't set minimum set what we describe when we talked about an apparatus which you could rotate space and a window and the apparatus that was 0 1 1 that logically the simplest thing you can do cost not the weighed corner carrots were discovered around just as a analyzing the experimental data that you we get by taking an electron Austrian orienting nearer detectors pressing me and bought the measure buttons and getting 1 nor minds former screen really is all over and over again accumulating statistics our is the army is supposed could get the reached physical really move into Lake City so much that they exist in this devastated and let the court if a capital for the reflected back it yet the but that have with jet they live trip but they also know that good question well and you don't ever from you have a photon or you don't have a 4 the radiation field itself is not in the state are of positive her whether whether or not you have a photon out there could also be thought of as a 0 or a 1 B radiation state itself the radiations system itself could be modeled by having another degree of freedom and the degree of freedom could also have a 0 0 1 0 0 would mean no photon present 1 would mean a photon president can't solve were happens but suppose we start with the singlet and triplet superposed in this way with no photon right to express the fact that there are no photons we have to put another degree of freedom and here only have to say this is tens of product with the state of the radiation field which has no 4 times keep in mind this year represents the 2 electrons either in the singlet state of the triplet state and this represents the state of the of the radiation field now have to wait for a little while and while the electrons are entangled here they are not entangled with the radiation field this is a simple product of a vacuum of photons of photons then a particular state of the year of the electron parents fact I said this was entangled is not handled this combination was just off them on a live off for a wild but what happens is the same stays the same and it does not emit a photon but the Triplette what happens so it it becomes the same way and it's a photon so what happens to the the state has evolved into linear combination of English times no photons and 1 for on another words it's a single electron at similar state of the electron pair but a linear superposition quantum linear superposition of no-fault torn and 1 for a single restrained but it's an important idea but you cannot only superposed they of a particle you could superposed the number of articles no particle and 1 particle could be quite the 18th 2 states convicts superposed stole the state of a system are vacuum and a space to stay with no photons stay with 1 photon stay would warm for common Yucatan them that's what's happening here you would eventually come to the state which is saying a lot but no photons plus 1 missed Mel Is it and tangled between DOS the electrons entangled with the a radiation field not because of pure product of electrons entangled each other yet singlet state only a single state appears here in a single state entangled state electrons are entangled with each other but they're not entangled with the radiation field on the other hand the radiation field is left the superposition estates of having a folk are not having for our soloists we started with a entangled state of electrons only on entangled with the radiation field I say this is on entangled with electrons because that's what we start with you'd be you'd be and no full-time start on entangled completely everything entangled waited for a photon be emitted when the photon was admitted saying the triplet state became the single state but we knew the photons and the result was an tangled state of electrons but on entangled with of the radiation broker pretty tricky set of ideas about but it was God knows how to lower numbers no nothing the cash or did not election flat interaction between the 2 electrons roughly speaking each electrons in the magnetic field of electrons but what they want to do was maximally and go line or better yet they want to minimize or make maximally negative the value of sigma about that's a limit a can't to do that God less suddenly started to move on to the problem of
particle motion forget particle spin who foreclosed real particles may have both position and spend but let's concentrate today on list questions relating to the position of particles the simplest example we can think of is a particle moving on a one-dimensional access X on before we study that I want to do a little mathematical and to a roast 10 minutes maybe less having to do with Fourier analysis Fourier transforms and this relationship to work to a Dirac delta function so let's stick a blackboard and do some mathematics are the funds illegally IPX said Peter B. number Piazza number now some of them will lead to the IPX at such combination of sine cosine functions and therefore a way of like character recall of a plane wave caught plane wave are just a simple way of course I'm signs a the combination of course I'm P X I'm so I Ciampi dialects and that figure was a wave of course is not move just right the IPX was given any kind of sort of move that of life apec on a very wide class of functions and certainly all of functions we will be interested in can be written ends some things Our plane wave or integrals Betty yet it the girls a plane waves and decomposing or function this case a function of X what's called side X decomposing it in the plane wave is called Fourier analysis that most of you know that but 1st to you some of you in any case not about Fourier analysis were would spell out now every function and that we will be interested In fact it's probably true that that excludes almost all function but we also exclude almost all functions when we say that their continuous when we say that differential we say we can express them in any meaningful sensible form we almost all functions but only take up functions to be reasonably continuous freeze on our tribe be very precise our continuous and they're not to blow up at infinity and the words to work through falters 0 minus infinity plus infinity almost all functions or all functions we will be interested in pot can be expanded in waves and this when are expanded means they can be written as integrals All P. What is Pete Pete tells you how fast the wave also this is some moderate value of a fight increase he likely away with very short wavelength decreased P talking about away with long wavelength is saying we can decompose of function in 2 waves of different wavelengths each wavelength has been a amplitude the amount of that wave that's in there I will call sigh with current eatery I P X so I could P is just amplitude that the wave of tight Pete has strained side P in the functions sigh of X for waste theorem is a theorem which tells you have a reconstruct strike appear with just definition is purely definition put a over square pie here that's that's a convention we could absorb it into site Twitter Pete cycle it'll typically is complex and so will sigh of X become a question both of them I intend for the moment become blacks now may happen that 1 of these is real and the other complex 2nd happened but at the moment we just interested in arbitrary complex functions that's right but the question is how do you reconstruct so I put look from X and I'm going to leave it as something you could look up if you don't know it it's a basic theorem of Fourier analysis that sigh twiddle of P In August the Siam next it isn't the growth of so I could look he is going to be an integral X DXL square of two-party times sigh of X natural thing would be to write theory to the IPX but not the minus side two-part thank you that's the basic theorem of Fourier analysis of cyan X is the Fourier transform of so could look of the find in this way the site of P is a Fourier Transform offside X very much the same pattern accepted put minus side we will need that so if you don't know that it's very important theory quantum mechanics and expresses a kind of reciprocal relationship between which of course ultimately are going to be position and momentum OK that's 1 little mathematical interlude that only the other has to do with the fortune and removing a combined together for a mathematician the definition of what we have that we have a a year into definition of adult a function at a high narrow functions so narrow with his essentially 0 and so high that the area under it is 1 we could call a representation of the belt function on a definition of the belt of fortune got the double function is not a real function infinite somewheres but but you're prepare reps but anal attitude about it did years from physicists incidentally to convince mathematicians it built the functions were OK he he let mathematicians them like them for a good reason that functions they can you can't square them or they just didn't square there still 0 everywhere except at the same point but instead of having a height the big enough to make aerial 1 them a height which is the square of Apple so they themselves are not always objects but they got over it and they go over a
program but mathematicians friends that it's OK to built a function OK properties of adult fortune the 1st thing in the way a mathematician would think about it is it's an operation that you could do a lot of functions in particular you can integrate any function of X any reasonable function of X with the delta function Delta our X minus X prime always let's go a step salutes you go do that yet let's take belt Bobbette's minors prime time's after 1st property of the delta function Is that says equal to 0 F X times Delta X minus X Prize why would that be true that certainly wouldn't be true for general function Bellcore theories and strollers biggest felt that is only nonzero when X equal X Prize so it's no big deal out of X minus X Prime was only 0 1 X equals X. Prime and therefore it doesn't matter what you put X for a time when multiplying it by a function so that that's property No. 1 now property the tool that pretty much follows from that Mel my Dec . 31 next property too into goal on Delta X minus X Prize is always equal to 1 that's just a statement the area under the bomb is equal to 1 and it doesn't matter where you pay for it but here over hearing here areas always equal to 1 of the integral of Delta of X minus X Prize but from minus infinity to infinity is always equal to 1 and the last property which follows from the east is that thing in the world the integral of dealt X minus next prime of X is equal to F L next time now see 2 ways 1 is that you could say Look delta function was only nonzero Amex's equal tax prime and soared picks out the value of X Prime letter acts or you could just I use property wears property No. 1 rewrite this in the former F of X Prize In the world Delta of X minus X Prize VX why am I allowed to bring me if elected prime money outside because doesn't turn elects its just now expired so From the point of view of the young as sergeant bring in money outside and then I say the integral under the function of 1 so I just get F X Prize but to those obvious properties of the built a fortune which show you might use Thaksin the replies were built a fortune there's a circuit now I'm going to prove a theorem which is provide AU representation of the delta function important representation of opera function love but I think we can erase this for the moment it ever probably didn't it to grow F as Dr. X minus X Prize is equal to FOX prime now this it actually is a sufficient definition of the delta function the way mathematicians defined as an operation that you duel on a function it gives the value the function at some point and this is enough to define our delta function it's not enough to define the function is only 1 function that if this is true for every function and only 1 function only 1 dealt that here that will fill at a property case of his commute taken as the defining function if all OF this is true this is what's at yacht BX BX could D X OK now what I wanted you is I want to take the tool copy equations and put them together he is a window on a right either X is equal to this saying but that the plumbing for this saying the 2nd equation and really express sorry over In terms of the functions side of exit self and see what we get but we will get something completely trivially stupid or will learn something in fact is we want something so it Rexon riders sigh of X is equal discover right now in the Gobi P. all square at 2 Ito the IPX and now I'm going to show in the 2nd equation for of sigh twiddle of and that is a bracket around DXC integral approach called what's you hold on a 2nd alone let's call Silex Prime and they decide next prime or we have to do is put a price on just change the name of the variable on the left side and to do so of course I have to change the name of the variable right here I don't confuse XNX prime time 2nd equation for site with of P that Icahn leave residues and just right there in the world be at Over square root Tupaiidae sigh X each of them minors I see X or just what sort through repeat no integrals are generally insensitive to which you do the integrals is particularly if their integrations reminders of the 20 affinity as they are he both of them if the well-defined and they converge in everything that it doesn't matter which order you do the integration so wet start that isolate something put here will be BP over to Tupaiidae but overtook fight we now have a single factor of 2 pi when coming from here coming from here but isolated Ito they are a key ex- prime minus X here veto the IPX prior to a minus IPX RTP X minus X Prize art and then we still have written down Siurek sought downside of yet coach a mail interchange order of integration of X P and right this Inter
will be backs into OBX that equal sigh that Brian so locally have we have a relationship between sigh of of X and 5 x prime we have a way of writing explored in terms of size of X but what does it but look what stands here take this whole thing here this is an integral part of a peach troubled but all it is function of X Prime Minister right but given with scored the capital built a fortune but call this a capital built a function right this D X capital Delta of ex- prime minus X this whole Messier according to pioneer everything else after integration Over P. Yukon of equals sigh of ex- prime where the tell you about this capital built the function
is just a old delta function there's just a girl Dirac delta function so if from been interesting relationship and 1 that is
at the heart of many many things in quantum mechanics don't peres into DP over took Piotr Ito they are a P of X murders X Prize is equal the Delta X minus prime level at kind screwball way of writing delta function by itself expressing it as a Fourier transform of or a sum of plane waves wicket also write this we can write were not writing the same thing running a different thing but just like he did changing Exon E by the changing XMP arrive mother equations D X Over each of the RIA X of P minus P. prime equals Delta off P minus P. prime this is these 2 are identical restore identical I've just change the name of Exeter P Peter X. this equation is true this equation is true because they exactly the same equation except in the changed the need to X with P & P with acts this is a this is an important decision fact built a fortune these are the basic mathematical fact that we need this year is not important for you that barely fell from a blackboard over there all they don't like the way I got the 2nd 1 from the first one but just changing PNX you go back to these equations and instead of eliminating site twiddle of heat eliminate acts by stuff substituting inside from here and you'll get there were are you just you just interchange PNX good right now we have what we need for the we will not testify where is pot add to it doesn't matter To be questioned it doesn't matter and the reason it is doped up X minus X prime I changed if you change XMX Prime functions of its argument you interchange Doctor of X minus X Prize seems Dr. that Prime Minister acts you could check that Soviet was doesn't matter of on where did the you probably right probably if I did the derivation of systematically next time next year but in the end it does not OK so let's or a this equation over here said they will be P. overtook pine beetle the P direct limits such priority 6 of equaled the delta of X minds Prime and into be X over Friday Ito ERA 8 PM latest people Prime tires X X is equal to 0 Delta P minus be prime side ballots comeback the world of a physics were mechanics of particle on white if you don't classical physics we would describe the particle underlying state of article on line by a point in phase space and face base consists of value of and the value of P 1 each for a one-dimensional particle just 1 1 x someone Pete whereby corner character how we describe the state of a corner chemical particle you could NATO guess on what I'm not how describe the state but what however we described and also normal basis of states that always is the question give me when you start in the system you ask what the the space of states and 1 way of defining the space of states is to provide an orthogonal normalized basis of space and also states that are linear superposition of that either of those possibilities for you might think so might wonder maybe V States candelabra Rep. Barbara positional particle X might think maybe it'll be represented by the momentum of a particle all you might wonder might states be represented as a pan and X and the peak this is large left drawing classical physics Pixar X and the people described position state part what is it acquire mechanics did so engine Kwan mechanics the X and PER too much information in the same way that for the spin knowing said that acts and the wife of simazine signal wire whatever it is too much information you can both of them simultaneously the same of course is truly club Canucks it's actually more familiar core of course the Heisenberg uncertainty principle but you can't know about those wound up over the other classical physics particle has and X end P and some sense in quantum mechanics of particle has hasn't ex or repeat meaning to say you don't try to describe it both described by 1 or the other posts will see what the relationship is moment we suppose that there is an idea that theory of an observable position but every value will be observable position this
vector represents a state in which the position of the particle is definite and at position xxx but it putin eigenvectors something which will think of an observable X growth will understand more mathematically a moment and like any good basis the point of a basis Mr. Right any wave function society or any state sector as a superposition of some of the bases factors as the basis vectors form a continuous range of possibilities we write they will be X replacing the solemn and then eigenvectors X kind something which depends on X this is completely analogous of what we've done in the past 2 if taken a arbitrary vector expanded in terms of eigenvectors of something and these are the coefficients here and we called these things the way function recall these things the way function I II and that they are given body in a product of the eigenvectors side eigenvectors acts with the state side that is so directs this is fully analogous to what we studied in simpler way function it is we projection of the state onto was stated that the next sigh wrecks and secondly sigh star of X time sigh Everex is a probability the park be found that position acts are strictly speaking we talked about last time strictly speaking you have to think of it as a particle as a probability density of probability per unit of probability per unit of OMX work you know your apparent workout 5 so that's a misses the basic framework of quantum mechanics just being repeated except for a continuous variable in a product between 2 vectors x next kind 0 unless equals X Prize and if these were continually these standard discrete variables we would say 0 my sex equals X Prize are 1 sex equals sex crimes right from the Delta X X time but that that's not helpful for a continuous variables the continuous variables we write the delta function the delta function direct function that's a continuum analog of the Chronicle Delta but where brought sectors in Kent vectors if they can't vector is described by a side next abroad of course would be described as similar way in terms of size star of X sell cats side next play functions roars star usual think nothing new here finally the inner product might finally theater product not between XMX current but between 2 arbitrary states let's call them and fight each 1 of which can be expanded by use of a wave function is described by a wave function 5 X brother the society could you tell difference between my size and modifies said yet misses Greek fight this is Greek side but I can never say Tsai I the Greeks going your test sigh if it the right that Barry that analysts but Korean this was described by size star X if them both appropriately and eigenvectors of X product isn't can't sectors and an Uzi in a product rule here for a the eigenvectors of Annex you will get of course at the inner product Is integral VX it follows by simply expanding plying expanding buying and using this product here but it's also hardly surprising the same kind of formula that we would have Our previously for discrete systems and last but there are questions but still as Adam and human but with death just he was based as friendly anything else is delivered yacht primogenitor knelt Doc Of the only hesitating is because we could imagine a particle moving to dimension in that case it's not clear yet whether we can only Ex Cold War new position and the white Cohen the momentum for example answers you can so armed but if I system was of particle in 1 dimension that X we are for a job to report chief that for
such a step it that is I get so that that is what we're going to 5 days said bilateral yes yes for Turner for right OK let's cellular phone 1 last the probability density is sigh star sign a downsized 5 West Side design integrates size Star sigh with respect X that would be the inner product of sigh with itself and that should be equal to 1 on the basis that the year October probability should be 1 another word nothing Milligan our physical status should be normalized normalized interval sigh size is equal to 1 OK now next last time we talked about some observables the position and the momentum so let's review a little bit about that let's invent and operate now capital X years maybe I should indicate that capital acts but for next line that makes it a capital acts and their foreign operator as opposed to just the number of argued that this is the operator or the observable which who was Eigen values on the positions of the particles observables observable values let's talk about action are sign of X when it hits me back there s or operators do they hit vectors and make I'm going to to tell you what the action by telling you how it behaves In terms of wave function instead of telling you how at summer abstract Victor I'm going to tell you what happens when they operate X it's always function wave functions and state vectors can be thought of as our room 1 being a representation of when x its side X what it does is as multiplied sigh of vexed by banks they said another words it gives you another function that's the nature of populated they act on functions to give other functions act on vectors to give other sectors if functions are representations are constructions which represent those vectors and where a linear operators are they should be such that they do something for the wave function in this issue what access to the wave function simply multiplies it backs it gives you know function with show same fortunes of different functions but but the bitter now let's throw something less true the eigenvectors values of this operator or what we expect them to be you of course do expect them to be something you may not realize it is moment that you expect them to be something that is so the city what they are you will say Oh yes we expected that OK so what it is likely to be the eigenvectors of capital X what it should be a steady that should be a state where the particle is localized had position X in particular let's let's talk about the particle be localized at point X what we expect the eigenvectors to be that should be of function which is 0 everywhere except that X not right that's what we expect that should be be concentrated attacks not so let's try the function Delta of X minus said not X knowledge of the planet he now it's a function of X the function of X which represents the particle at position x North and let's ask the question when you'll operate the operate X on this wave function where you get fully and is huge is multiplied by lax but now we go back to 1 of the properties of the delta function they but when you multiply it by any function of X Yukon replaced by a function of X Prize Seoul warriors case sex not misses is also legal text not trans delta of X minus that's not boat is equation this is exactly the equation for eigenvectors on the island Eigen value of the operator X operator X it's the wave function and do the number of times the same way functions operate its way function gives number of times with function it's clear that this is an ideal vector of the operator With Eigen value x not quite we can make it a list of eigenvectors of areas various observables starting West eigenvalues and eigenvectors the eigenvectors associated with a particle localized point text north that is equal the delta function of X minus sex exactly as you expect right for brokers for should amassed finish of OPEC's were about about momentum again momentum being an observer won't you could observe it hits you side of ahead you feel it back he recoiled bans were murmured so the momentum means the same thing quite characters doesn't classical mechanics at conserved and or few absorb something with some
momentum you you require right where the momentum operator the momentum operator a justification for this From later are we did talk about it but I wonder back momentum was core they operate a momentum now momentum is equal to his No . beating around the bush works they work to go it is equal minus I'm willing to put in some units here later on we will set each bar equal to 1 which is a choice of units but for the moment let's keep age barbaric times the derivative operated a D cynical deal less time for the operator be last time we studied whether capital D was lesion or not we found there was not but if we multiplied by I either with a plus and minus sign the matter if we multiplied by an IED then it becomes mission so if we do we worked out last time if you're a little trick with integration by parts hurry found D was not her mission but I can be part of this is the definition of a momentum operator or it's a bunch symbols so far but was P do when it acts on a wave function just gives I H bar times by DX so is also an operator it's also an operation operator it's a linear operator and it is our mission linear operator PT so that's what it takes to qualify as observer with quantum mechanics bomb at this point since I is only a function of 1 variable it doesn't matter whether I used partial derivatives or just ordinary direct Steve what's called ordinary derivatives IDX next question was the eigenvectors of the momentum the eigenvectors of the momentum satisfy P onside or minus I side by DX equal summer I value some observable
value us on possible outcome of the experimental torpedo French side this of course be not as a number Eigen value our weekend what's just freak get rid of the miners I hear it becomes a plus sign on the side of candelabra each bar so this is just the equation for an exponential derivative being proportional same function bath solutions facade that unique apart From a multiplicative constant side is eigenvectors Laura as a function of X and let's label it by tried just a label the fact that eigenvectors momentum p it's a function of X more resilient to the RIA peanut acts Austin wrote thank you for it undertaking V-8 forum for about 10 seconds I just want to point out that the figures
a waiver like things away like playing signed co-starring and has a of allowing the way is the distance that you have to move so that the thing in the exponent changes by 2 pot lures of recall the wavelength lambda P although each bar Times lender equals 2 pilots a definition of the wavelength how far you have to move so that the finding cosigned returned themselves or that the wavelength is tool height age bar divided by P. not peel being the momentum of the particle so here we see the 1st example of the relationship between momentum and wavelength larger the momentum the small wavelength of course we have not yet justified the choice of the terminology momentum from this thing but there are we can keep in mind that later wrong will go back and see that this really does behave like momentum but so far it's just an abstract observable we don't what it means OK Sergio Ayala Land physical this and now that I have demonstrated that let's get rid of age bar because forget about anyway but get rid of age Bora lose formula here say this it is now not clear yet but normalized is correctly not clear that this is normalized 1 let's take meet in a prop up between 2 momentum states like this is missus here is a representation of the momentum state P not for but butterscotch P. persist coach Pete Carroll North the answering any real purpose here but thank appeared just Pete let's take it a product of told momentum states P prime and what is a formula for products is over here and here it is over here once I start fight so that means in the product was integral father each of miners I keep my ex that's the wave function of peak prime or complex conjugate of it and the wave function for Ito the X DX awkwardly into eat the key minus P. prime XBX now does things right P is supposed to be a observable mission operator and the eigenvectors of her mission operators should be orthogonal and this should be 0 of P is not equal to be prime it's pretty clear that is 0 0 is not equal to toupee promise for the following reasons the real and imaginary parts Ito the IP be people X signs and signed if he is not equal to P prime is up signs and signs of P minus P. Prime X call signs and signs are as positive as they are negative view integrate them over a large interval you get 0 as long as you pick up an integer number wavelengths the report so as Luz P is not equal to be prime this it will just oscillate self 0 an hour equally positive and equally negative for both the real and imaginary parts so it is true they Prime is not equal to ask P back theater product is equal to 0 he is not equipped comprise what about P is equal to pee prime interval from minus infinity we are looking at the other black for their work there were firmly say about stuff of peanut from Pete equal to P
Prime this could be the right angle of 1 integrator spaces infant cell and find his son looking smelling like it is the sergeant smelling like it is function in when P is equal to pee Prime and 0 otherwise good things but we don't know what the actual numerical coefficient but yes we do appear we have our and they will wear is that this 1 already at Inter will be X all took is dealt the repeal be prime the upshot of this is just that in order for this to be normalized so it in a product is a real delta function you have to describe the divided by the square root of 2 young square root of 2 big deal square root of 2 party this is the wave function for a particle of momentum text pioneer a particle momentum p we now know what away functions of particles of momentum that would be about a Jew did you try again so but no 0 it is it my head volume would rolled out just test 0 yeah this is a bad is a guy for news summer rooms the answer is that you really justify body starting with the fury on a interval U.S. you think questions about convergence of the interval In total doesn't really converge so you really have to to go back to some earlier starting point which has started with teaching quantum mechanics and a graduate class of something I would have started by thinking of fury of a finite double or even better on a fine at periodic intervals worked it out and then taking carefully the limit as you let things get better go to infinity and then Everything could be justified with readers we don't have the time and so I don't do that but of course for each itself right now that's a fact about the Fourier Transform but I did see I did she at some point I think that changed and X integration with a P integration and 1 of those integrations didn't converge did basically the same cheat that I did hear from but it but I I wouldn't cheat you if I didn't know that the steps can be filled in and if I didn't know that if you go and find yourself a book on Fourier transforms you won't be able overstepped yourself so far I haven't really cheated done was just a little bit of a short a short cut by now we've talked about the wave functions of particles with momentum p but no wonder thought about a different concept the wave functions In the momentum basis so far all these wave functions have been thought of as functions of X P just a parameter the momentum of the particle about Norway function is a function that's all incidentally notice something important bro something port the wave functions of particles with definite momentum oscillate endlessly forever and ever and absolutely on local bodies there is deal localized as it could be a killer signs that side of X is uniformly spread out over the entire relaxes in no sense is being positioned well-defined uncertain as it could be on the other hand the momentum is very precise eigenvectors momentum operator if this were the wave functional a new residents momentum it would be Pete measure position it would be random over Texas very much well much worse than the very much like releasing the X moves EU representations I'll wear 1 be definite at the cost of the other 1 being as indefinite as possible cases just 1 minus warned here much worse sold when a particle is localized momentum it's completely localized position we will be able to say we want or I would like to do the reciprocal thing also like to go near the direction for that we want to do is construct a notion called the momentum space wave function the momentum space wave function is related to the probability amplitude for finding different momenta Alexei we can construct that about her thick with initial here but sir but if but we defined in general we define a wave function by taking the state vector and projecting it on eigenvectors of various observables and that gives us the wave function in the bases associated with without observable are here projected onto an data banks that Disney the wave functions side if instead I projected onto a state of definite momentum that gives me something that could call the wave function even the moment the basics it's not a function of X so it doesn't look like a wave in space from overweight and space by it's called the wafer and a function of momentum for his given name what's called sigh twiddle of P just this I argues the twill notation just to remind you that 2 different things at different different animal from the year from the ex-wife function How we calculate their so let's suppose we know 0 B X representation of the of the
state vector Siurek how we calculate well we take in a product of these state vector side would be better Pete but we know what that is as the Inter will be X I think I've lost uh only the only but only equation Moses sci-fi it was an integral the size star of 5 banks but yet books OK so I want be a product between he warned of the wave function of P and position basis it's just what is it eat of I P X over the square root of 2 friend that's what we have so wrote emphasized he of acts of the sky over here and then we have a multiply that by sigh of X and integrate but that's exactly equal the cycle of Pete up over the same notation that I use of Fourier Transform another words the momentum state wave functions strike will of Pete is exactly the Fourier transform Of Thee position space wave function that's the upshot momentum and position are related in the same way that ordinary variables positions and are Fourier variables later OK said it would have cost me and if you know what this says is if you know the way function the position basis you know everything that's enough in fact could cover a wave function in the momentum basis and also the probability of fine particle with different moment after all was a proper abuse of patients should be a capital size should be a small size wave function but across the probability From measure given value momentum is just a side twiddle start of pity times sigh twiddle of some way or another you know what the wave function of particle ways sigh of acts but did not make interest in measuring acts you're interested measuring peak how determine the various probabilities for measuring different you take that X wave function you subjected the Fourier transform you find the Fourier transform of it and it's square the probability of the moment is is extremely similar to going from leasing Z bases the Cigna X basis go through it them CEO kind narrow find the similarities between 2 or more b all other than the pay was able to 0 minus RIA Beebe X and you apply this Tour positions based wave function you apply positions place where function defined back but but but et not quite Parra not minus ID by DP plus idea by the my assigned different has a great deal to do with this minus assigned here political work it out but with Brooke back at 3 this is the place that for Isenberg here and physicians and diets isn't it but the time and energy use of are time is not usually regard operators are not usually regarded although you could armed don't you do measure annually to watch the hand of your watch is observable and story Clark readings could be thought of as observables something a little bit funny about them and Oregon now but Burke usually inquiry Kenneth Kimes just considered the parameter by which state all but is true if you have a clerk for example and that you're interested in making that they're making an extremely accurate observation Clark will be at the cost of a lack of precision the energy of a cork gap is here is a collective magnetic field just almost anything you could think of will have a year earlier conjugate variable electric and magnetic field but no temperature and pressure statistical mechanical variables McCormick chemical her without good yacht I will I'll take angular momentum and angle the angular momentum axial so actually axially symmetric system the angular momentum and angular orientation would be another now OK there are lots of others usually for example Johnson field typically the field and it's derivatives Our conjugate variables through all kinds of New across yes but it seems that would promote a discrete states observed states to continue with this kind of change representation felt that met faces states the operators operated on the way function the lower but we could
right but we we could also reprimand also represent the operators as matrices and the cat as columns a column was just a list of values are an amplitude as a function Weirich flying how I as a function of the wagon value of really is I think this is closely related to the representation of operators as matrices just concrete concrete action In a particular basis when you're right column vectors you write them in particular basics cry last remark arrest remark about uncertainty is the uncertainty principle we for this particular example here are not going to derive the uncertainty principle market they may get to it but it really is reversible thing is simply the statement that about fish it's a statement about Fourier transforms Furcal and transcends Kwan mechanics of can't mechanics any time you have a pair of quantities which are related by Fourier Transform in this way is generally the case that the narrower 1 Fourier pair for half a Fourier is the brought the other 1 another words the narrower range of key the broader sigh of X will be advised versa so I don't go into a mathematical statement about about the uncertainty principle here but whatever it is it really does does have to do with these character of Fourier our pairs X P in this case will come back to it but I want to get on it was the Hamiltonian particles just a simple list possible Tony into particles and how they govern acquire mechanical evolution remember what a Hamiltonian Mr. Hamiltonian is things which 1st of all enters into the Schroedinger equation and which also governs the kind evolution of the state vector the Hamiltonian this led fight the classical IV of energy just as it has in classical mechanics if you go back we him discussed this earlier but a purpose corn starting point was the idea of a unitary evolution of state vectors of state vectors evolved a unitary transformation we said well what happens if you make a pie you know you kind tiny little change in time that correspond to some sort of differential change the state vector we found out that that differential change was governed by her mission operate a call the Hamiltonian sorry if you are forgotten the details of that go back toward Mount Lebanon right now Turner but you remember the equation the equation for the evolution of state vectors as I but expires in for the moment figure Madigan who 1 of these side by DT the way the state vectors because writers terms of state vectors state vector is generally are function of time state changes with time you could take the difference between sectors and you could differentiate vectors with respect to primer for vector happens to be time-dependent took a differentiated lying beside is equal to the Hamiltonian operator acting on strike that's the kind of dependent Schroedinger equation we work that way I think we worked out an example for a speedy processing and a magnetic field where we just used for the Hamiltonian 1 of the components of the spin here is a general form of the Schroedinger the time-dependent Schroedinger equation a let's try out from simple Hamiltonian Stewart kind of things we learn from them and I think we're just study war 2 and will go on metal at the very simplest Hamiltonian that I can write their own which is quite illustrators Milbury a variety of around wherever are everywhere age no ageism operated a Moses operator its collision operator h those get permission the Hamiltonian numerous we're try each it is equal to a number the constant C for constant finds the momentum thing but see what does that Stewart kind of thing we learned from on is this 80 who reasonable Hamiltonian for particle young actually it is and will find out the moment what kind of particle but that's not the usual thing what we usually right wrote a part be squared to win the momentum squared over with twice the rest will come back to that but this is simple this is simple and worth exploring 1st just to see how the apparatus works everything here should be capitalized everything Castillo operators OK but how how is represented in terms of wave function in terms of wave functions and wooed away from society sex wave functions in the X bases the moment the bases functions the X-Base just projections onto the X eigenvectors or we can do that inside equations here on what this reads then is RIA bar of X by misses the way the wave function changes is equal no H terms please sort of see that and he is money Miss I each bar Dubai DX was on minus an are in H and the Debye DX sigh of X there were now I've rewritten this
equation in terms of wave functions the of concrete representation by Everex and I've used here the assumption that the Hamiltonian is just equal to see finds the momentum and the momentum is on side age for Beebe but this is a very and of course I is not only a function of X will sign a function of time as a function of time because general state changes with time that's all OK we can cancel out each Boris and I just get these signed by C from minus city planners decide by ID X or decide by D. C plus decide by IDX a city is equal to 0 that's a pretty simple equation and effective so simple that I know of what not only what a piece work a solution I know all of the solutions of this equation a but or the solutions of this equation on the general solution a function I X minus CKD many function restlessly because like any function which is a function of X minus the words it doesn't depend on extant T separately in only depends on the combination X minus we will solve this equation of our differentiate with respect key that will be seat and the derivative with respect X capsules equations any function Of this form will solve the Schroedinger equation so let's think about a functions formed was due for the look like they start at equals 0 0 causality here With any functions sigh of acts of course we don't take any function I want my functions to be such that the total probability is equal to 1 of the in of size our size should be fine equal 1 on a recent blow should be if should falter 0 my sleaze can integrate them but they and wealthy the whatever you like some sort of lump it can oscillate oscillate armed a rabbit general now elects tea right now we let forward what happens to solution what happens solution is its babies exactly the same shape except every feature of it moves with velocity features other move further right with uniform velocity function is just a function which is moves rigidly without changing its shape because it's a complex function as a real partner imaginary part both of them maintain exactly the same shape and they just slide down the axis uniformly with a velocity c Knight Josie a because light this is a photon well this is not really the correct description of 4 times of pretty close to the right prescription of neutrinos at least neutrinos which will with the speed of light Real neutrinos probably most slower than the speed of light but there are on measurably so this would be of darn good description of 1 one-dimensional neutrinos X said that we would have they another possibility that they know is what all of that pure right moving the tree the pure right and they don't have a funny thing that energy could be part with a combat is not the square at the energy can be both positive and negative P E is a thing which could be positive or negative so this particle has the property that can have positive Levangie but that's not here is a picture of the wave function moving down the axis with uniform will also be the whole probability distribution rigidly moves down the axis what happened the expectation value of X but Moses so the expectation value of X lawsuits and the best quote Canucks of the simple system I would just point out that free compare this with the classical mechanics of exactly the same system Hamiltonian is equal to CP compared to how we do the classical physics use Hamilton's equations so here was mainly for this reason right here that we do all of classical mechanics last quarter of the world not just this 1 but for all sorts of things are on Hamilton's equations for further Hamilton's equations Hamilton's equations are H by D P equals X . and H. by D X equals minus P . that created by is is just equal to see right engaged by DX that's equal to 0 so what this says classically is that the momentum as conserved there will come back for the conservation of momentum in quantum mechanics but it also says the position of the particle moves with velocity C and quantum mechanics the whole probability distribution in this case this is a very very simple case of engineered to stab a simple answer the whole probability distribution moves with a birth In particular the expectation value of X moved velocity the expectation value of the position behaves according to a classical
equations of motion now that that this as I said this is a very easy example which is a little too simple to be perfectly general but there are both those illustrate the year so this would be the Schroedinger equation for this very very simple Hamiltonian inquiry about a charter school more and more what about us and of course this would make this a relativistic particles would not be in nonrelativistic Oracle was a guy with the speed of light Young a life with feel pressure for article no particle which is always constrained and move with the same Milosevic so right right so for example no just photons Our choir of electromagnetic field sound waves and crystals also have quanta sound waves a cough Oman and released at low energies of phone lines move for the uniform velocity the uniform velocity being the speed of sound so if you had a one-dimensional full nuns you would describe in a similar way to throw prototype for 3 days not electrons don't the generally with the speed of light is a lawyer no no no no no you with this this a lot of physics for this particular kind of particles was no matter how you throw it with whatever momentum you throw away it's always move the same way I want the big a lot of places they lost velocity being means momentum each quarter but quite peace a P squarely in but that's close to the right as Klaus arrived at a will for a relativistic particle of course he's not reveal and bath you're on the right track there is a position movies said no no no no history of the not quite right arm only very very simple situation because the wave function with this rigid behavior and maintain shape not true or more generally figure nonrelativistic electron workouts equation our are these shape of its wave function is not maintain so while more of those things so are gradually falls apart disperses diffuses disbursed recall right I picked warned just because it's easy to solve but firm but doesn't represent Vermont general kind of motion of a particle prior to its list of about sticky guests with sticky Guestrin relativistic particle I think I'm ready earlier showed you that these equations of motion that expectation values clap quantum mechanically on very closely related to to the corresponding equations of motion classically Hamilton's equations apostle before bracket former sold you could imagine that if you wanna represent the quantum mechanics of a system which is which whose classical physics you know you basically using Hamiltonian now we will come back combat justified better but the natural Hamiltonian to guests for from nonrelativistic particle just girls T-square over 2 and this is not a particle with a force I made this the article would not force on a free particle energy kinetic energy he is growth one-half mass times velocity squared
momentum is em times V can rewrite the kinetic energy if that's all there is of course each as peace squared off 2 and that's the Hamiltonian furry classical particle noticed this Hamiltonian has a funny feature particles could only move to the right no left moving motion all we moved to the right path and safari fame asymmetry between left and right has to do with the fact that he goes from my his PC when you invert left and right the Hamiltonian changes signed in Hamiltonian that that's a funny thing you have the energy of a particle with negative momentum is negative energy of a particle positive momentum as positive that's a little bit weird our money laundering nonrelativistic particle energy is independent which way it's moving and that's insured by the fact that the energy is proportional to pee squared not so sewage start with the energy being P square over to win and work out the Schroedinger equation for it now we will work be working out to genuine Warren Schroeder a original version of the showing equation for a free particle Friday ages eagled be square over to an H is also i.e. Debye now puts brakes bar just a terrorist arms show you would've done it on site that's the left-hand side thought of as an operator operating on the Great a each for Dubre duty and the right here right hands are recorded losers H acting on sorry but a just peace Cuero to so that is 1 over to and there was a key whatever P is we have to square it but let's write downward His 1st minus H Mart Heidi I age derivative with respect to x but at but the square feet square square meters means multiplying it by itself so course square this I each squared and then the square the 1st derivative is 2nd derivative you supplier twice and size on acts as it that's a strutting their equation for a on ordinary nonrelativistic particle moving on a one-dimensional on I was blessed with a pH forest RICH bodies Robert D. C physical Tom minus each bar coed minus comes from minus squared auditorium these 2nd society by DX square Is it kind a wave equation incidentally this is also a wave equation is a wave equation and sigh is a wave function misses a wave equation size also away function is ensuring their equation for ordinary nonrelativistic particle if has the property that waves of different wavelength move with different velocities will I'm getting a little tired store or were equipped but waves of different half Waverly move with different velocity because of that the wave function does not maintain that shape factor cancer 4 pardoned and a spread with truncheons spreads out with time unlike the very simple case over here in this case no matter what the wavelength no matter what the momentum is they almost the same velocity not Sonia different momentum with different velocities and so wave with a given the shaky different parts of it will move would differ velocity different there we've laid different wavelengths it it will move with the velocity and I will fall also revealed more detailed behavior booked on for more please visit us
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