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The Theoretical Minimum | Lecture 10

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beyond the material chorus but let's just talk about the model we talk about a Hamiltonian energy which is thirsty can they could be positive or negative the wave functions forgiven Peter adjusting the IPX and has nothing better about each of the last IPX or P being positive or negative that both that perfectly acceptable plane waves but 1 of them has positive energy in 1 of them has negative the reading here now negative energy it is not such a good the in around in the real world negative energy is not such a good thing real world because if they work particles of negative energy the world would be unstable the fact would not be an empty space would not be the state of lowest energy you could take empty space which we did say has 0 energy and then start putting in particles of negative energy particles of negative energy will lower the energy and anything that will always the energy tends to be favored America lots and lots more and more particles of more and more negative energy are you have to get some cemented positive energy from someplace that thieves positive energy from someplace but you could for example radiate photons saying time produce more and more of these negative energy particles Castillo particles of negative energy empty space would just be unstable unstable with respect to creating more and more positive and negative energy particles you can't you have to conserve energy so you have to do both but that we have is the only positive energy particles by contrast is no way that energy conservation will allow you to make them if the only positive energy particles of the vacuum has 0 energy its lowest energy state that nothing can happen to it to create articles so good stable world requires all particles have positive energy relative to the vacuum is not good stable world OK so what do you do about it while it was Jirak who understood what to do about it but only in the case of affirming arms family nuns I simply particles that cannot be in the same state 2 of them cannot be in the same state as a rule where that rule came from it comes from relativistic quantum field theory but it's the same as the Cali exclusion principle if a node chemistry you know that the 2 electrons in an atom can be in the same state and you have to fill the Adam with particles of different energies different states and the same is generally true particles that are not that participate in Nepali exclusion principle of called Meaghan's quite a call for me and I don't know I mean they should called it's obvious they should be called power arms and particles which is don't participate in exclusion principle called both on but of course properly they should be called Einstein on the was 1st study never mind firmly arms bars on firmly answer can't be put in the same state bars on canceled Barak said this look Ivory simple solution of this does the magic that the vacuum empty space is really a state in which all the particles of negative energy are present simultaneously filling up the backyard you simply can't put any more because not all out of particles in the same state but everything else is empty In fact if you would say that think about from Olmert let's assume that the real vacuum is a state Our absolute lowest energy absolute lowest energy now take absolute empty space which is think of ordinarily empty space is that the state of lowest energy no I could put 1 particle of negative energy and quality of energy stocks Orrin G. whatever state say I will find another negative energy particle the put into it and Laurette's energy the only way can have the state of absolute lowest energy is the simply filet fall from of all the particles which had negative energy in other words all the particles with negative momentum sounds crazy but turned out to be right but you can only do that with particles which have the property that you can't put more than 1 of them in the same state the promoter the same state it would be no end to your ability to keep dumping negative energy particles in the backyard so this is Eddie reasonable period said Barak periods of very simple basic baby version of the Dirac equations what is and it's OK theory for neutrinos of one-dimensional neutrinos is not a good theory for our times times or bows on and you simply can't play the same game if they were negative energy bows on sucrose simply dump them in the same state more and more and more of them simply destabilize the world so I know this is not a good period bows arms into the period but that's taking us way beyond this course the course material with card and sends a lot about the foundations of quantum mechanics and very simple quantum mechanical systems with covered although most nothing about what we could call the applications require mechanics of money the applications of quantum mechanics but studying quantum mechanical systems elite it even get the harmonic oscillator that's a disgrace but we're not finish of quantum mechanics next quarter would only do relativity field theory but they regard the have to come back quantum mechanics of motor studies more about the base sticker L structures that appearance can solve its so embarrassing to teach a whole course corner mechanics in vigor to the harmonic oscillator but that's where we are you could ask for your money back on at the same table get a catalog From the ideas of entanglement that expect there such a respond
particularly the use of some of was extremely interesting involved some of the year traveling back and forth on the Internet I actually did follow and sometimes it was generally good and sometimes the excellent so I got some some sense of satisfaction of teaching them parliament move on a little bit today about a stirring equation not the generals running a report of a special strutting requires the Schroeder actually wrote Maryland the connection with the things we learned up till now was the wave function and the wave function is simply the amplitude that a particle is located at a particular position words it's been a product of whatever the state vector happens to be with state of definite position and also writes a momentum space wave function the momentum space wave function just be projection of the same state whatever happens to to be harmed a momentum item state eigenvectors the momentum eigenvectors and video the position eigenvectors or the rather of momentum wave functions position with function are related and related by Fourier transform that's basic structure of position and momentum ever particle has more than 1 direction from moving in for example suppose a commotion EXY playing then you characterize me positional particle by to court next the Bhutto coordinates commute you Ken measure both the X and Y position of a particle simultaneously now that's something of follows automatically from what we set up the little now number all that's really an empirical fact it's an empirical fact the structure of the quantum mechanics of particles only conforms to allow us to expand if we assume that the different coordinates the different directions the of a particle moving New that means that the wave function is really a function of a single variable but a function of a collection of variables the collection of variables being the position of various coordinates of the particle if we have several particles then the wave function becomes a function Of the coordinates of all the particles when I gotta go into that very they're just tell you that the wave function of so I love the way function of a collection of particles it is not a function of three-dimensional space it's a function of all of the coordinates of all of the particles in the same way that phase space the phase space of a particle is described by a collection of a large number of coordinates end lament that in the case of classical physics the wave function of particle is a collection of the core of our system is a collection of his on a function of a collection of all of the poor so for example of that tend to the 23rd articles side would be a function of Canton 20 3rd quarter cell that's our that's the the basic structure and if you could go a momentum space by Fourier Transform and ride the wave functions instead as a wave function of all of the particles you could even have makes situations mixed representations will you may describe a wave function in terms of the accord of some other particles and many of the other part is also out there with their focus on the case of one-dimensional motion again with back-to-back just discussed the behavior of the wave function properties as the little bit mainly properties of assuring a equation 1st the uncertainty principle where the uncertainty principle come from it comes from the fact that X coordinate and Pete don't commute whatever took or 2 observables don't commuted becomes impossible to measure both of them simultaneously because there aren't eigenvectors which eigenvectors of both observables in particular the eigenvectors of X are extremely different every eigenvectors of X is extremely different from every eigenvectors Pete the eigenvectors X sharp narrow highly localized functions based and the eigenvectors of momentum 1st of all complex Peter the IPX but they real and imaginary parts are oscillations or space service was based on our enough space for supper particle Phil's face it's just that the article is everywhere equally likely be everywhere momentum is not so that's the most primitive observation about the uncertainty principle if you know the position of article you certainly don't know what's momentum particle is in a position against state it's very far from momentum I state the momentum items it's very far from position I stayed off that's the most primitive version of it the same qualitatively more qualitatively and they will come to quantitative from quantitative precise the moment but if we look Eds the wave function as a function of X and its relationship by Fourier transform these P of the square root side twiddle the I P X and I won't bother writing the other relationship of the relationship is the 1 that rights sigh of P in terms of size of X and just started fooling around trying to construct sigh of PEO sigh of X 1 from the other by Fourier transform you very quickly discovered that the narrower the distribution in X the broader the distribution In P will be that's a property just of the Fourier analysis that the narrower sigh of
Exeter's the broader society is and vise versa sign he was very narrow it means but it just basically a single plane wave of Missouri broader necks and so that's a pattern of Fourier transform the narrower a function is the broader its Fourier Transform that's the qualitative understanding of the uncertainty principle but let's try to do a little bit better but strive to see if we can find the notion of uncertainty the Pacific Prototheria amid a pro a slightly simplified version of the era mostly because there's just too much on the blackboard approved general fear food a simplified version of it but the simplified version of it contains a basic ingredients but 1st of all what is meant by the uncertainty in a variable you have some distribution that variable and the 1st thing to do before discussing the uncertainty in it is shift the ax sees solid that average of experts say this is the 6 so that the average of x 0 you could always do that if the average of X is not 0 4 you have more function like back at CERCLA 0 over here if you put your center of course it's over he had obviously the average of Exie expectation value of X will be negative you put your coordinates over here Media the average of X will clearly be passes and as you move the origin who eventually find the position where the average X is exactly equal to 0 so let's simplify our story by shifting course as a justice of the court heard shifting coordinates on the average value of X is equal to 0 average of annexed it was EUR what do with average letters equal you're what the uncertainty uncertainty by definition is average of X squatter in looking at the average of at cost it's 0 because anything on this side is balanced by something on the side of positive over he Exs negative over here looking at the average of it the average Joe will cancel with you choose your right but the average of X school acts were measured relative so this located axis he will not be 0 x squared is not 0 here x squared is not here the fact that square the only place where that's where 0 was right at the origin and so there's no way that the average of index squared be can be 0 the fact the brought the wave functions the larger the expectation value of X squared will be clear so a good measure of good measure of with of the distribution been called uncertainty in X and this definition really of the uncertainty in X is the average of X squared the average value of X squared with a given probability distributions given that the average of X itself was equal to 0 that is the answer certain Pemex has just called Delp x squared just got positive numbers squared and the broader distribution the larger will be less value calculated if you know the way functions very simply take me into goal of size star sign X sets the probability of fine particle X kind X square that's the expectation value of X square a C average erect square probability defined the particle position next client X squared that's the definition of the square of the uncertainty the square of the uncertainty of X Wednesday its definition but good definition theory writer appear gulp XML square people the Interpol of Tri-Star side context where acts but now were that is certainty in momentum to the uncertainty and momentum as the firing a very similar way 0 1 4 1 4 this is equal sign X squares it expectation value in the state of X squared thought I was a quantum mechanical quantum mechanical formula the expectation value of X square assumes where about the uncertainty in in momentum and fact for that matter what about the uncertainty in anything the uncertainty in anything assuming its average is equal to 0 is just the average of the square quite that general definition incidentally if you'll love you don't want to shift the coordinates so that the average is equal to 0 0 you could use another definition the other definition is that it's the expectation value of X squared minus the expectation value of X square the 2 different think the expectation value of the square of something is not the square of the expectation in particular if the expectation value of X is equal to 0 it does not follow the expectation of exquisite 0 This is a more general definitions of the square of the uncertainty in X but as I said you could always shifter act cities around so that this is 0 just choose as 1 here of your about the expectation or about the uncertainty of momentum the uncertainty of momentum as defined in essentially the same way you can define it if you like
using the Fourier transform the wave function and that it would just be an are going to use this definition but let's just do it about the peace accord that would be the integral society twiddle of sigh of relief funds peace square feet the probability that a particles momentum peak times P squared integrated overpaid but I'm not going to do it this way that villagers observed this quantity over here nothing but fight the square outside the to work In the position representation we could do that because we know what the operator P when on a wave function side that was Pete du what does the operator P dual when acts on a wave function In India expertise what does for 1st of all whatever you take expectation values while in a product you having to BXT from abroad that they hear you have star acts the cat outside of this is squared I'm was be squared P Is minors on Dubai DX using now but momentum operator PT Miners I EX and peace squared just might be set by IDX where 2nd derivative of Queiroz rack twice with Pete wants them quite strong after this is equal to the interval the X signs of X these 2nd sigh by D X squared with a minor sign the minus sign coming from what minus are white times minors on May look at this new warrior bet on the Simitis signed he alchemy How can uncertainty in she's been negative wasn't mean that have been negative uncertain and the uncertainty in Haiti signed here what's going on what's going is the integral in here is negative that's old going army into a and how do you see that negative you integrate by parts just remind you the rule for integrating but part is if you have users overall functions effigies and of OF DG by dx dx physical Tom minus Indigo be athletic times G. VAX this is but as long as effigy go to 0 appropriately at infinity so boundary terms don't contributed integration by parts but look at just look the structure of because we use it over and over you could switch the derivative from 1 factor to the other factors at the cost of mine Pepsi therefore they OK so let this is a 2nd derivative 2nd derivative is also of course a 1st derivative namely its the 1st derivative of the 1st I've IBX lets us do integration by parts all we have to do is switch the debate the exercise started and change the minus side OK so this becomes X for the VAX over here bebe IDX who switched to the size star at the cost of the changes signed becomes plus and this becomes these sites now look older have here we have designed but EX we have besides start here this factor is a complex conjugate this factor adjust complex conjugate each other what happens if you multiply complex conjugate function by compote conjure restores real and 2nd Sakamoto positive so be it the grand in here is positive sign that changed this most definitely a positive interval here so square This is the expectation value P squared if there is a lot we can always shift until it is 0 now the real questions could you have brought the average of X and B average peak with 0 that's not so obvious isn't I'll are you could always you could always shift 1st to shift Ex equals 0 and then you could shift the peoples 0 Our visit trick you know I think from wireless combat me again and I want you to track the answer is yes you can take a wave function and shifted both In X P until both of them have average equal to 0 will come back row . 1 in a while but I suppose we've done that and let's see if we can find AD relationship between the uncertainty in X and the uncertainty in the mountains and the relationship but I suppose it called a relationship of mathematics but inequalities of equality
quality should be such that 1 adult Texas small built that P is vague when Dr. P a small Exs big or better yet that the product of built X times dealt the P cannot be smaller what then aged Potter now I've lost barring here I'm lost its bar because of 70 1 of 4 put it back later but if you wanna put in here and now we should probably put here is 1 of the age square here are square he square here but let's not Brown right place yes but let's work with its particle 1 and later on with finished off put back a year of the dimensional quantities OK so here we are what tools do we have to move constructed equality is basically 1 Brazilian inequalities in mathematics but the cover most famous as useful and the only 1 that I know that a particularly useful it is a triangle equality our Harry triangle inequality some of those people thought appellate and we did it comes in 2 forms a statement about triangles of all things what it says is a triangle that sum of any 2 sides is bigger than the roadside obviously true right some of this size and the side awhile that because the shortest distance between any 2 points is a straight line so the shortest distance between these 2 points as this 1 and the other 2 sizes some of the other 2 sides is certainly bigger so we have 3 sides state is also another version of it they thank got prove it 4 of the other version but did not the have a version very well also goes want a let's close for B E and went scoreless for OK so armed 8 plus B is seat or 8 plus B has put bigger and bigger size OK 8 Class B is also were the same argument say is bigger than 8 plot speed after these are vectors think of that is a matter that any user vector 5 years on the display on if crew C is not papers be used a being in this case I might think our persuade B E C K still a triangle a Class B is bigger than the city which is the same as saying is bigger than they they bombed let's square at rats squared and this becomes a square square Riverside Ave was be squared plus Y C B is bigger than this guy over here but what's the square of 8 plus beat it is a squared Class B squared plus while easter . product of a beach where they get that from I wrote that 8 plus B squared is just 8 plus b . 8 that's a square of overweight record and that's a square both be squared was twice did not Roddick the alien and so we have now been inequality from which we can you race a square and be square from both sides here's another form of a triangle inequality works erased the Toulouse what it says is if you have 82 vectors given 82 vectors but because of our warm ADB but the product all of a sudden away from 1 of them forms a link the other is always vigor because to disappear as bigger than the knot product there's another way to think about that what is it that product in terms of a the of later be and something else were in need another calculated product cosine of the angle so a dot B is of course nothing but may be a cosigner the angle because I'm an angle is always less than 1 so the triangle inequality is pretty trivial simply says that the magnitude of the product of a magnitude of 2 vectors is always bigger than the dot product between them because the bar product always has a course I know this is true for vectors in tow dimensions reader mentions a 100 zillion dimensions it's even true for vectors complex vector spaces it's just generally true of vectors in any kind of vector space that magnitude defined as the square root of the in the product of
a vector with itself but this inequality Cherlyn factor but services right a property and let's go where it because we're going use in the square formed a square be squared is greater than a peaceful just squared girl I haven't done anything and that's that's the inequality that will you dear yes it should be great equal to the only equal His when 18 Berenson election as a great memory Karzai figures equal to a cyclical that's our that's going to be on tricks Olympic venues more welcome back With Tom Yellin role T an awk so promising we have dealt the X squared hiya soda like a squared where ever built the Peace Corps adheres sort of like be squared now we want to show the bigger something so the trick it is to defined the right age and be used in terms of size and exit and they just use here but I'm going to make 1 simplifying assumption already made to simplify assumptions that the expectation value of X P 0 that I will show you how to arrange and that there's no trick the simplifying assumption are going to make only because they don't want to fill up the blackboard with more algebra need necessary Mr. something you could sit down with at home and go through it the more general case I'm going to assume that the wave functions sigh is Rio if I did we be chasing around sizes stars and I would eventually yet tighter you out and you would probably lose the threat of the argument so I'm going to assume that sigh Israel will save us some algebra it's not general but it's very easy to fill in the steps for complex I'd be easy but it takes a few more lines and I just I just after the sitting down and doing it on a piece of paper I just decided it was too much for a blackboard
presentation to do that is to do it for the simple case society is OK so here we are we we can get rid starters and end on figure lost an equation here using the fact that he is minus ID by DX we proved that this is equal to integral these sites star IDX these I'd like to remind you of quick wit arrived we get it P Square was the 2nd derivative well my 2nd derivative we didn't integration by parts take 1 of the group over here that Dr. P. squared all the fat probe is the product of this time this is is bigger than something product this friends is bigger than some OK and to do so it's just it's just a trick now think of AT&T inspectors and thanking the figure may be as the brought sectors are kept sectors society is real it doesn't matter whether you have Robert there's a cat vectors but he is be the 1 that maybe I'm writing and now in victory Brock kept a notation triangle equality is perfectly true for Bronson cats as well as for art directors wrecked at 80 there's going to be described as gonna be factor who's wave function is not sigh of X but X 8 Is the vector whose wave functions His exercise Ex-Im words its X operating on side and b going to be the vector P time sorry for basically minus I decide idea when I write equals 0 meter say that aid is the vector whose wave function sigh of X B is the vector whose wave function it is desired IPX apply the triangle equality that OK notice that with this definition here Help X squared is a square look at In a product of a with itself a squared is just the interval of side times X squared so this Is this year a square with this notation or about this want this 1 here is this century b In
a product of the B with itself How come I don't have a reminders I debugged DX here if I put my this ID baby X over here however minus minus sigh which is minus 1 how come I don't get Oajaca might have a plus 1 of my public services be squared that's a square in a square times be squared it is also dealt the X squared times Delta P square not felt that squared built the P squared a squared be squared Be triangle inequality is literally talking about OPEC's where the P. square so now let's look at the other side of the inequality here the other side of the inequality is a dot B squared so let's computer 8 . be it see another product of these 2 sectors and there was square that we take its absolute value square because we take its absolute value I don't have to keep track of II minus signs in the end will go away because they think the absolute value and so the inner product of a dot B is going to be equal escargot that have an integral all 1 no 1 thing I did say that I'm going to choose site to be real so all of these sites stars don't matter decide every place start you could think of Sighs OK so let's take 8 . 2 square that's going to be a which is sigh of X X X 2 3 written the office of order and then times B E which is going to be decided by the X as I said I'm not worried about signs or is for the simple reason that is going to take the absolute value that was play a that will play no role and what sort of this quantity now we have a dot be squared so I could multiply this by itself from 1 the multiply itself bias suffered from multiplied by another in the glow of the same height wiII be side by BY DX is you want bundling is writing the same thing twice so if I can evaluate this I'm also evaluating the OK so he is the trek the trick Is this a be signed by the extra inside the ring a bell to you sigh buddy extends no formal don't know no the lock literal life was dominated the early on right track them it's a derivative of the square of outside into 1 half the derivative of the square of sigh a derivative of the square outside is equal to twice society decided so up part of the factor of 2 0 what we have here it is and 6 times the derivative of size squared with respect to Exeter ability that yell out of you don't OK so I can remove this I replaces size and at the same time a factor of 2 0 1 half that of the same thing with the other same thing yeah Our sigh squared while another factor to which makes us all together a factor of water and as I said address this is just the original 1 doing it but should just write square but it just look the square by writing the same thing down again OK next integration by parts to have the written here at the Riverside Square but still just a derivative there when you have it a function times a derivative to be integrated you can switch the derivative the other function at the cost of a sign a really there's no cost him sign because is another sign from this 1 over here and so was not real cost signed by side taking guarantees be the absolute value so equal to form In duo of squared times DX by DX and likewise over here in science squared BY but you want the X fight exits sets from earlier as is warm and this is just squaring same thing we are finished were finished because what is this the goal of one-quarter to get caught with the interval size sigh squares I start side size Rio size sizes square what seemed sigh square 1 what is seen the role of the probability of wall space so each of these integrals is 1 if the wave functions normalize we assumed that the wave function normalized if we assume that state vectors normalize this whole thing is just 1 quarter and we have proved that the uncertainty principle over from the triangle inequality Graf dealt X squared times please squared is bigger then one-quarter one-quarter I remind you came from 2 factors like this or if we wish to just go we can just right this is dealt with square-rigger both sides of the ex Delta P is bigger than 1 one-half now what about are Unix X and PEX are not in encouraged to each other so the product of an and peace can be a pure number was immersed to annex is appeal over age pop song really this is Delta P already each bar put back the units were basically dealt ex Delta P is bigger half but this is the fear research the uncertainty principle that the uncertainty in position and the uncertainty in momentum as bigger H bar now this would be true for any direction or space pick any direction they take or any particle any coordinate generalized coordinator mechanics its uncertainty Pansy uncertainty in the corresponding conjugate momentum is bigger than age that's a bigger NATO for over 2 hours 5 any questions about about this particular version of the arms the which the both the teachers flows and deal with a lot of course at G was viewed my now to integrate this equation all right this time to integrate 2 questioned the integral I'll have tried Jean costly integral on duty crimes X is equal to the integral of Debye IDX that but the interval of a derivative is just a difference between the Inter granted To win point of integration services between NATO and . infinity and my innocence and symbol stands Fred G. evaluated the affinity minus FG evaluated the other in the the integration that while means is that the side 0
infinity which would have to to do with it normalized did the goes finite probability distributions normalized it means that you wave function has of 0 so this is your and this is just a roulette you can interchange where you put the derivative at the cost of such OK other question about the uncertainty principle not it it all not head-on don't sell small when you prove that something is bigger than something else you convey a question can ever be equal something else and the have knowing they have that a quality which was not the high distant equality we could approach could somebody come along smarter endeavor promoted a tighter inequality that it's gotta be bigger than something else stronger inequality and this case is not sure what you can find wave functions for which dealt backstop the peas equal each bar over to the minimal uncertainty way back over to its it's not for me for the last time low EI it does but the circumstances under which you use it tho less clear and I don't really have time to explain how we make it at some point in these costs will probably want to think about it but I don't remember that the time dependence Kuwait From the time dependence on sigh of tea is a function of time is a somewhat over the eigenvectors the energy of scores from of all the energy eigenvectors of each of them minors by each T. at 5 times the energy eigenvectors times a year at times a year and a function call a squat fiery so the relationship between the wave function as a function of time and the wave function as a function of energy is once again use some not things and energy with phases like To get things is a function of time that's the Fourier Transform relationship so it's on but exactly how you use it to to give you an example of how you use that you have some wave function with past your mowing past you and you ask how long does it take for the wave function actually passed you and this Kabila light been incident but how long will it take for the wave function to move past you another words what's the uncertainty in the time at which the wave functions might have Max been maximum long take maximum forget past you Vatican called delta T uncertainty in time the wave function arrived right you know odd related to the uncertainty in energy of the wave function we bigger the uncertainty in the energy well together with the small of the uncertainty in the energy the bigger the uncertainty in when that wave packet will pass you for example that's an example of the energy time uncertainty relation sold Eddie pair of things which are related by Fourier expansion and this is a expansion threatened society as a somewhat or integral over oscillations things which are related in that fashion will have to uncertainty their lots of other conjugate pairs like this the angle of supposing you have a rigid body rotating about point where angle data characterizing its position then the angles data and the angular momentum have exactly the same kind of relationship uncertainty in angled kind of certainty in angular momentum is also age for us take world who also very similar kind think any other questions Paris but tender literary earthquake crew the firm that places a wave the equation for the way a wave function changes with time it also governs the expectation values novel wave function changes with time you could try the deduce how expectation values of observables change with time 1 of things we would like to do is to see the the relationship between the motion of wave packets governed by the Schroedinger equation and their classical counterparts namely the motion of system systems in this case being a particle will like to see that as long as the wave packet is narrow and not to crazily spread out the cure for tonight's bell-shaped curve back state is a bell-shaped curve forever for some link time that but the way functional the center of the wave function moves in a way which is familiar from classical McCann we need to be able to do that make any relationship between the classical and quantum mechanical things over talking about OK now we talked about this summer river let let's 1st right there the general throwing equation on the take you back a couple of lectures now piggyback cup of lectures and then go through will be derivation of how always Hackett more the answer is the wave packet moves so the center satisfies a classical equations of motion are not going to do that in a new way to go right Back to the old ways that we've already talked about and to do that 1st all in order to know how the wave function changes we need to know the Hamilton and all that surges remind us of works the way you were away state vectors change with time I decide is governed by the Hamiltonian acting on size if we let it take for Hamiltonian Peace Corps aware that seal Hamiltonian for classical particle why we should take it required a chemical particles and what what might wheat add to that to put some of but the particle of force field might end up potential energy function potential energy function because of that so that would be the classical Hamiltonian for particle moving in a potential energy function of X and we can take that we can lift that right from classical mechanics and think of it as a quantum mechanical Hamiltonian and then check afterward is the wave packet really moving away classical mechanics would say that the year of the classical particle more but so let's try that we what's we will just work will while just work with this abstract equation here but let's just Stewart says a Osup a chance at a instead of working with abstract kept vectors we can work with wave functions with this equation would you would be I saw he is equal now peace quarter was PEP is my society by DX P squared minus Deeb IDX squared on 2nd IDX squared was a 1 auto and minus 1 or 2 and I'm sorry for what is real Exs via the phone operator yes Pemex acts as an operator by definition is a definition I told you where the
operator X . the uploaded X multiplies the wave functions just multiplies it by X the corresponding thing for a general backs as the meaning of the operated via X is that multiplies the wave function by we'll talk about that much we didn't talk about functions of X assumptions Alexa justifying the obvious way as a his operators they operate on wave functions just 2 multiplied by the you yah yah functions of operators yes BOX is a function of position x multiplies sire view Rex also more hours and you can't write this definition on when you make a definition of course you later proved that it was a useful definition just make definition to make definitions and show that they they either are valuable because the easy to work with for the valuable because they relate things to previous things you already know blogger want shot was that with this definition of how sex acts that the Schroedinger equation will in fact make particles move according to classical orbits or at least make wave packets so far is it don't spread too much move according to be classical equations of motion self that's why literary sacked and 2nd derivative of site and the other just plus possibly next time Scitex when via the XIX on Silex it is multiplies it this becomes the Schroedinger equation this it is the Schroedinger equation how the wave function changes with time also depending on time let's at the same time right no I wanna do this I'm gonna be interested in decide by 18 so I think I'll multiplier away this where we do we multiplied through I get rid of I will multiply by a minus write letters items eyes what multiply by minus sign gurus minus sign and I hear and put my life he that's it sigh . if you like the left inside a side not lets also right down star of ex ante a With respect to see that's not no equation it's just a complex country of this 1 and that 1 will be minus fight over 2 m these 2nd five-star by IDX squared plus ii objects psyop and not yet as a starter in various sectors a your in the old guard sigh star that write those are showing their equations very close to each other I just wrote them both the I need a multi but now what we wanna see how various expectation values change with time if we identify the expectation value with more or less the center of the wave packet of a wave packet looks like there's the center of a wave packet will be closed to the expectation value can so let's identified 5 the general Lurie packet with the expectation value and ask how expectation values change with time Serie work it out that's our next Wednesday of and calculate debate on the expectation value of tanks and that of course Is Deeb IDT of the integral sigh style of x x squared not just X Siurek New Year's start to give it a good velocity of the center of the wave packet that elicit it's the velocity of the center of a wave packet white does change it changes because sigh change going on outside changes we are outside changes because we know the Schroedinger equation so they get a little exercise a little a little bit too tedious might case that the dawn black that sniper tedious but still alive Mejia make comparable pieces the fixed shortcuts but there I'll tell you what I OK so visible we wanna calculate and this course Exs at Texas not dependent on time but sigh stance sign dependent on time so I differentiate aggregate terms From each 1 of these and this will give me in will be star of ours is riding a bike star X society plus a star X so I got at 1 time coming from differentiating with respect to times I start another 1 from differentiating site each 1 has accident and now of course this is a sad about 1st thing is we don't really have to calculate both of these is a complex conduits of each other they have sigh star staff here we have so I thought he signed so I start here and X Israel so in fact all we have to do is calculate 1 of them an its complex conjugate so we have to do or equivalently just take twice the real part adding something to its complex conjugate the imaginary parts canceled just twice the real part we can use that Russia needed but but but twice the real part of it prices and keep in mind that we throw away anything that's imaginary With throwing anything that's imaginary because we have the complex conjugate that will throw away OK so now what plug-in size star by BT over here and that gives us ought to them these 2nd scsi were worry plus some salary plus I it's the upper equation his 2nd exporter minus 5 8 X society now the 1st thing we can see and immediately that the term of via the X is pure imaginary sigh started his ex society sigh star side is certainly Rio X Israel via VOX potential energy Israel and is a nightmare so the 1st thing to conclude is that the V X doesn't contribute at all it doesn't contribute all because when you multiple you and the complex conjugate you canceled so in this fact in this equation we could throw this away sigh start X V of exercise that's 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evidence of all eyes and minuses let's try this 1 5 P B average of P is got by acting with P sorry and that some miners I give us a minus of probity and then Debye IDX aside by the ethics we could but that they sigh but we could put the derivative over here by integration by parts and assigned changed but who Louisville over here cooking a and we would a heart attack on the where we do not we wanted time deliver the moment from what was once they wanted the time during the momentum I was governed frightens equation P . is equal before that was forests in terms of the things I've written on the blackboard my this TV X so it's ourselves minus by acts both solid in the form of that deem IDT of the expectation value the momentum is minus the expectation value of DVD by TV by X it is also a function of X B is a function of X Devi buyers by the X is a function of X as call the force for says is the average rate of change of momentum is the average of the 4 were the 1st to be reviewed by DX is my so that's a lot of shall we last show that Pete IDT His DVD X but and how we once again with a short by using the Schroedinger equation that Cal us how if Howell things change with time OK so let's Sweden now on they get a little bit unpleasant what's so much right off what I'm debating pilots what start working around them at some point they say I know this goes away this doesn't go away but we will see our orchestra 1st of all this disaster integral sign star 5 kinds decide IDX minors ii and then there's another turn that come from that came from differentiating size start with respect to time there's gonna be another termed where by will hit side peso that will be again and minus into star now we offer of differentiation doesn't matter so this to be written as our by D X of site got I'm just wondering a 1 . regret not having done integration by parts but all matter because we do the integration by parts and will be good I will regret it was just go ahead 0 extra time no I don't think so but I don't think so no I think it's right OK after you know what I think it will be convenient to integrate this 1 by parts integrate this 1 by-products I will get a class by here and DEC I star IDX here another reason that's convenient is because of things in here is the complex country thing here OK side that started well but the complex you sigh not by the excess the complex conjugate of but not always it's I times the different where I get when I take off my 8 times the difference of toe complex from you get a real thing again right Seoul bomb I think it shows the real part Our this well it's just keep going my eyes but just keep or we can erase this and remembered in the Ind that the answer has to be twice the real part of this twice the real part of laughter keep track of both of them OK now where we don't reply in the Schroedinger equation and pray like meant that all goes well OK so I've got is equal told over to areas these 2nd sigh by DX squares minus fight 3 of X site this II cancels assigned do a minus signs of this I was and I hear this summary minus the Iowa he attends EIA over here I think will make plus plus the case we have a somewhat unpleasant Tussing present tree Moon Cabrera distaste I believe is pure imaginary it is it is used this is pure magic here the way you see here the way you see here is again integrate by parts of the PC fight integrate this Peace but parts would get a derivative will get a turn with a single derivative of society and a 2nd derivative of size drink will discover is basically the difference between a thing
and its complex conjugate be a minus sign OK let's with such a severe doubts it if I differentiate or sorry integrate by parts that'll bring after the derivative over to here it's pure imaginary I know is whenever you have AT quantity complex conjugated plans a derivative of young conjugated thing you integrated always pure magic reviews that trip before I'll just tell you right now this piece is pure magic sucker the b of any significance the significant piece is Togo sigh stars like act via back side that's all that's left after we take into account worthy of the little intricate details of complex conjugation the attacks and we I remember whether we double the girl not yet come I think we doubled I think we double yacht the idea with Beverly we have to double this yes we have double art and a torch complex conjugate right talks complex if I could do that right now but a added to its complex conjugate plus the Indigo offside X B sigh of at least 5 idea I have taken this complex conjugated it I've taken this complex conjugated Israel OK now almost there we have these sigh Star Of XTX prime society which collaborator X 4 where VOX X times LSD sideways this thing here so I can't decide star next course I start and decide Mr. size but I respect X is misses derivatives I start franchise precise stark terms a derivative of side that is nothing about the derivatives with respect X costs sigh start side tourist put that in end of the next step be to rebut porch In by parts don't think that there is a derivative of the year and I will give me my yes the integral offsides kindness DUX or effect but Ciara here right over there said the integral Ala sigh Saipan DVD X that's the expectation value I love DVD X this sort of thing is the X minus the expectation value of DVD X number words it's exactly so if found that a us like the painful Our integration by parts of a little bit of algebra throwing away some things which I pulled you a pure imaginary which they were refined the Peabody DTE is exactly equal to minus speedy baby expert In this sense of expectation in the sense of the expectation no while I say they'll on this this also was exact I didn't do anything illegal here know any approximation of February use or where to where may be important that the wave packet have and I shake the expectation value of TV bite X is not the same as V R D but just veto sister take beat the expectation value of b of X not let's call it after sex for the expectation value of effort back which is after all miners Stevie but X the expectation value of a function of X is not the same thing as a function of the expectation value of this represents the expectation value of X the center of the wave pattern and this is a function of the center of a wave packet it is not the same thing is the thing we calculated in general this is the expectation value of path of acts when I give you an example would a very very different with video they could be extremely different suppose for example that a half were equal to X squared supposing the force happened to be X squared and supposing the wave packet consisted of a pair of bombs but is our warned that's what want yes that is exactly where 1 was centered about 0 was the expectation value of X what's it F Of the expectation value of X 0 0 and on the other hand was the expectation value of X squared so not 0 this 1 has neck squared the equal to the X where there's non-zero sold when wave packets on right not nice single bomb which are mainly characterized by this centers like that in general you could not right the time rate of change of momentum it is equal to about forests evaluated at the expectation value of X it's only if the wave functions are nice and that concentrated or a fairly narrow range
at the concentrator were reasonably narrow range then the expectation value of f of X it the same as the excess f Of the expectation value of this is where we have roh a cheated a little bit and saying this looks like a classical equation emotion that depends on the way packet being nice incoherent and well localized weather circumstances under which away function will remain nicely localized I tell you what the circumstances are the circumstances are that the particle is heavy particle was heavy end fix to settings of particle being fairly heavy and potential energy not having To Mideast bite about having spikes or something like that spikes on the wave function hits bikes it tends the break up for example you have some sort of wave function coming in a nice way wave packet coming in moving to the right and ahead of point structures what would do it will spread out all over the place that way functional disintegrate all over the place if on the other hand it hits a very smooth potential of some sort it will go through the smooth potential moving more or less according to the classical equations of motion so we we don't expect quantum mechanics to reproduce classical mechanics in every possible sock we expected to reproduce quite mechanics circumstances where church where particles a heavy and where potentials are nice and smooth and don't cause away from Chen's break-up pieces of a degree of scatter along the place OK on the any of arms and in the award None of America we don't have time for it to be a place for his 1st bad well OK the situation which is tends to be were unknown when don't win the potential has structures and features which lets they have some societies associated with them structure retreat called back size associated with structure here and we're dealt the a significantly smaller than uncertainty in the position of particles if the structures of beyond where we were colored features sharp features of the wave function take place on a scale which is much more than the then the size of the incoming wave packet then it will break up the year away functioned pieces each 1 will scatter gross Kadyrov different dimensions of different directions after that the young window that the city collapse of it is in those cases were but were just another judge expect some of you should but of course right when they're now there are few basically when the teachers and the potential are shorter waiver are shorter than the wavelength of the year of the particle no break it up and who are now if you want to take a year the bowling ball and you would ask what is don't X well but typically at stroke that typically are dealt the peak I'm stop X rule is bigger than age far lend any reasonable cases it's of water each for now P that's about as concentrated as you can get it but from ordinary acroscopic object the uncertainty principle was pretty well saturated felt the peak times the OPEC's is more or less of article age barbed wire that Seoul is a very complicated question but a very complicated dealt P got the P is mass terms about the velocity so the uncertainty in velocity times uncertainty in position as age following him Nevada bowling ball down on the ground you know very well that the uncertainty in velocity is not very big the uncertainty velocity is in particular as a gets heavier and heavier you might expect in the velocity it's more smaller but in any case the in down here after and so well dealt the as and gets small as and get smaller and smaller Delta X will get bigger and bigger and Delta text will get bigger and bigger and in particular it will tend to get bigger than the features In the year In the potential sonar highly quantum mechanical limit where is very small and dealt the X tends to be big wave function will move under the influence a ragged potential with chair which as being much sharper and much more featured in the wave function itself that's what breaks out on the other hand gets very large dealt text get small dog backs will tend to get small as it is and it's very large and Stauffer a large bowling ball the wave packet might be very very concentrated moves through here in those like a plane wave function and the kind wave function thinks that these features are very very broad movie from broad smooth features doesn't disrupt the wave function and break it up into pieces sold large and smooth potentials is the limit of classical physics light mess an abrupt potentials is more corner care young viewers are small enough you know what it is a letter from Lorna you need a F-15 this place the pendulum that if
you take a very very moved potential of the kind that you might make let's say it be a couple of capacity Place capacitor plates separated by a meter Ralph premier a centimeter where 1st small electric field between now and then the electron will move through it a nice coherent almost classical article on the other hand if you take a potential associated with video with the core of the of the Adam nucleus that has a sharp featuring and that sharp features will affirm electron wave comes and hits it was scattered all over the place so if the interplay between the man and the sharpness of the year of the potential this time it actually have fully not it is that but that those votes out of both the problem breezy the classic experiment was done in 1911 indicted electrons lose them with alpha-particles alpha-particles coming and hit a gold nucleus gold nucleus is a small thing and the result was that the goal of the players when alpha-particle comes in ways for mechanical wave of alpha-particles comes in it's that kind eagled the glistening scattered directions that's why Rutherford was a bit surprised to see the CIA's alpha-particles get scattered right back at him whether Taiwan that tell them that the nucleus was very small meant it just have been about articles some of which are at the center of some of which went through here by where fact we really had was waived in the way of Canion 50 at the kind of gold Nucleus Inc of scattered all over the place and with electrons varies at 11 accelerators when electrons it is small targets they just get out of the photons door also funds the war also electromagnetic waves if they hit structured objects from the structure smaller size small in the wavelength figure scattered all over the the place if they had very smooth things they tend to propagate according to the geometric objects for example if you have a piece of material with a position varying go index of refraction if that index of refraction is very very smoothly and very slowly over the wavelength of via photon then geometric optics is a good approximation a funny other hand you have a diffraction grating of distant comparable us more of the way want them to review should the diffraction grating about all reporters not just for electrons or charms four-pronged pace of a formal request from rural home for mercy from review how formal or please visit us
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Metadaten

Formale Metadaten

Titel The Theoretical Minimum | Lecture 10
Serientitel Lecture Collection | The Theoretical Minimum: Quantum Mechanics
Teil 10
Anzahl der Teile 10
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen.
DOI 10.5446/15010
Herausgeber Stanford University
Erscheinungsjahr 2012
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Physik
Abstract (March 19, 2012) Leonard Susskind concludes the course by wrapping up the major concepts that were covered throughout the quarter and discussing some of the limits of the field of quantum physics. In this course world renowned physicist, Leonard Susskind, dives into the fundamentals of classical mechanics and quantum physics. He discovers the link between the two branches of physics and ultimately shows how quantum mechanics grew out of the classical structure.

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