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Lecture 04. Complementarity, Quantum Encryption and the Schrödinger Equation


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Title Lecture 04. Complementarity, Quantum Encryption and the Schrödinger Equation
Alternative Title Lecture 04. Quantum Principles: Complementarity, Quantum Encryption and the Schrödinger Equation
Title of Series Chemistry 131A: Quantum Principles
Part Number 4
Number of Parts 28
Author Shaka, Athan J.
License CC Attribution - ShareAlike 4.0 International:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor and the work or content is shared also in adapted form only under the conditions of this license.
DOI 10.5446/18882
Publisher University of California Irvine (UCI)
Release Date 2014
Language English

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Subject Area Chemistry
Abstract UCI Chem 131A Quantum Principles (Winter 2014) Instructor: A.J. Shaka, Ph.D This course provides an introduction to quantum mechanics and principles of quantum chemistry with applications to nuclear motions and the electronic structure of the hydrogen atom. It also examines the Schrödinger equation and study how it describes the behavior of very light particles, the quantum description of rotating and vibrating molecules is compared to the classical description, and the quantum description of the electronic structure of atoms is studied. Index of Topics: 0:00:20 Localized Wavfunctions 0:11:30 Fourier Series 0:13:21 Quantum cryptography 0:28:32 Time Evolution 0:47:22 A Free Particle
welcome back to the 4th lecturers can chemistry 31 today were going to talk about complementarity quantum encryption and the Schrödinger equation 1st of all I suppose that we actually know something about a particle but not everything so we know that a particle maybe localized or in the vicinity of some region of space and will stick with one-dimensional problems for simplicity here so we have a variable X and we know that the particle is probably at a position around X not the question is what should the wave function look like for a particle like that and if we assume a real wave function we can write something that creates a peak near a In particular point X not in terms of a galaxy and function of function that small and comes up smoothly and down and that's a very nice function has a very simple analytical form which I've written here and it's characterized by a position where the PTS which in this case is 6 not and buy a standard deviation which has to do with how wide the distribution is around the position acts not in other words how peak the wave function is and how closely we know the position of the particle so sick my this formula measures the width of the peak distribution and a small value of Sigma means that the wave function is peak more strongly it's really very tall and a large value of Sigma means that the wave function while it has the same average value where the peak is it's much wider In fact I've plotted here the way functions for segment equals 2 and Sigma equals 1 and both the wave functions when you square them and integrate them have unit probability of particle is somewhere in the universe but you can see in terms of this craft that it's very likely that the particle is quite near acts not that's the most likely place and if Sigma's too there is some likelihood that the particle could be 1 or 2 units away and then by 4 dies off quite quickly buffer said mn equals 1 there is 99 per cent chance that the particle is going to be within 2 units of the position x not but we think the particle is this OK and then so the particle is much more localized forcing me equals 1 in this lets us have a tunable parameter we can use the same formula we can put in different values of Sigma get a family of wave functions and that we can analyze how they behave as we move forward and tried to discerned things about momentum ,comma the uncertainty in the position of the the particle now we previously worked out that the momentum operator was minus side each body by DX and we found the Eigen functions for the momentum operator there the complex exponential so I mentioned that they corkscrew 1 way or corkscrew another way depending on whether the mo momentum is positive in the particles is moving to the positive x-direction or the particle could have negative momentum and be moving in the negative effects direction but the size of this corkscrew does not change in space it's completely uniform everywhere and so in a momentum I can function we know absolutely nothing about the position the question is if we know something about the position but not perfect knowledge we have a distribution like discussing and function then how much if anything do we know about the momentum and are the 2 things related to each other and to find that out of what we have to do is we have to figure out how to write this smooth Gallician function in position as a linear combination of momentum Eigen functions and then the coefficients of those momentum Eigen functions will tell us how what is the chance if we make a measurement of momentum rather than position that we will get a certain value for the momentum eigenvalues Of course if we make a measurement of momentum we will have changed the way function in a very fundamental way and so it will not have the same distribution and it may not be located anywhere near x not after that and it turns out that you can kind of see what's going to happen just by imagining this then wave function if I have a thin wave function and I have a bunch of corkscrews let's just forget about the imaginary part for the time being just plot the real part if I have a bunch of functions that are going up and down and up and down and I wanna make something that's quite narrow and then pretty much 0 outside I can't just use a very long wavelength because that'll never actually be very tight and I can't just use 1 wave Conseco's everywhere that's never going to be 0 so what I have to do in order to make it work is I have to take a bunch of different momentum Eigen functions and they the art they have to at least oscillators fastest this thing is
dropping in other words if I signed way that has to drop at least it that fast and know what has to happen outside is they have to all kind of interfere they're all still there but when you add them up they cancel out 2 0 which is of course the beauty of waves and which is what light does all the time it may take many past but many of them Council to 0 and so what we anticipate in this analysis is that the more localized the wave function is in position the larger the spread of momentum Eigen functions were going to have to use because we're going to have to use something that oscillates quickly and that means it has a big value of and that means the total spread can be quite wide and so what that means if we mention the momentum is that we're going to get wider distribution of moment of life the here are a couple of momentum Eigen functions of the contribute To this Sigma equals 1 way functions and you can see from the graft but the amounts of these momentum Icahn functions are all small the people 0 is a flat that's 0 momentum as just flat no corkscrew and all peoples 1 is a very lazy thing that's barely changing and 10 is going quite quickly but because the function dies at around 2 P E equals
10 is not going to be enough momentum to getaways that oscillates quickly enough to actually get this peak to be that narrow and so we anticipate that we're going to need more waves than that and because all these values small and because there are a lot of waves what that means is we get a very wide distribution of momentum functions and that's because we need this wide distribution to build up this localized position calcium iron calcium position function what I've done here in the next slide show what happens if we take our original Gallician wave functions and we approximated bye 20 momentum Eigen values and we choose the coefficients so that we get the best fit between our approximations and the true functions and away you can see is 2 things there are are not so good here the 1st is that because people's 20 is not fast enough we can't make the peak narrow enough so we can bring in the spirit of the peak quickly enough because we don't have anything that's dropping that the 2nd thing that's wrong is that we get
these wiggles that keep going outside of the gets smaller and smaller but they go outside the region that we want and need this is reminiscent of diffraction of light through a hole and many other kinds of problems that encountered quite often and it is just a fundamental property of trying to cast this function in terms of the signs and cosigns or equivalently in terms of complex exponential How can we cure this well instead of using 20 momentum item functions suppose I use 50 and I tried the same thing that we get this grass here and now it's far far better now it pretty much tracks the GAO functions there are few Wiggles World was going to have a few wiggles unless we use an infinite number of functions the few wiggles outside and the peak doesn't quite if you look closely at doesn't quite get all the way up to the top of the Gallician and function in the center and that's that means that we're missing a few functions and the functions were actually missing are the ones that are also creating these wiggles outside if we put in these extra functions from 50 to infinity and
they would be very small amounts toward the end what we would find is that we could match this many this calcium distribution absolutely perfectly and we can do that with any set of bike and functions we could use position I can functions Wikinews momentum Meighan functions Wikinews Eigen functions of some other operator and it because the Eigen functions former basis just like any point in a 2 D plane can be written as this much Jackson this much wine and there's no escaping if we use enough of these Icahn functions of any operator we can exactly match any kind of way function were going to encounter and then when we look at the amounts of these basis functions that's when we find out what a measurement of momentum will give us for a measurement of any other In fact any reasonable functions and wave functions are always reasonable because we have to be able to differentiate them and integrate them and so forth any function at all can be cast as the sum of signs and signs or equivalently as complex exponential speeded the Bader and that is actually the principle of Fourier series expansion and that is another good math subject to study so that you understand exactly how this type of thing works In fact if you do study that in "quotation mark proper course in mathematics what you will come to the conclusion is that the uncertainty principle can be seen From this aspect as a consequence Of the position and momentum Meighan functions being related by a 48 transformed and that's right away gives us the uncertainty principle and makes a quantitative of we have a very narrow distribution in position we have a very wide distribution momentum we need those fast momentum might functions to pull the skirts and if we have a very narrow distribution and momentum that means that the corkscrews go way out all over the place before they actually interfere in goal was and that means that we have a very wide distribution in position the particle can be anywhere and they're just the flip side of each other but we can't have them both being narrow it's not possible to use 1 corkscrew and make 1 spike in position because they're totally different things when 1 is narrow and the other is what now this measurement as I said if we actually measure that the momentum or we make a measurement on the in wave functions we will have changed it because it is not an item function of what we're measuring and as with all the worry about security and privacy I thought I would do a little bit of a topic here called quantum cryptography using quantum mechanics it's possible to make an unbreakable code that nobody can spy on you and in fact it's a very simple an ingenious thing it's being used now in some places in the United States and in other countries and it's based on the observation that if we make a measurement that the wave function must fall into an item functions of the measured variable and that means if a spy makes a measurement on something that we're transmitting the spiral but influence the data and we can pick that up so we can know something's wrong we can know somebody spying as well and we can then just simply stop talking for so how does this work well but we need some sort of a thing to descend some particle and the easiest particle to send as a photon we can send a photon down a fiber-optic lines we can send a photon through free space lasers are very efficient at making photons that have very nice properties the best properties available very narrow wavelength range and and so forth and so on and we can send lots of photons 1 at a time very quickly and that allows us to send a lot of information and and to the establish of method of communication How could you do this well the 1st realization of this scheme was 1st put forward as far as I know but by Bennett Benidorm Posada given the reference here and slide 107 in 1984 so quantum cryptography has not existed for all that long and this scheme is called 84 after the 2 authors and the year in which it was invented there are other schemes as well because once you realize how to do it there's a lot of ways to skin a cat as they say but I just want to talk about Bibi 84 and show you how it is possible to establish a communication link but and encryption strategy where nobody can find out what you're doing In this field there just like in quantum mechanics we have sigh and we always use Cypher an unknown wave function that we're going to find out about a week 10 use 5 4 basis functions in this field of communications the 2 parties who are trying to communicate our are always Alice and Bob by convention and Alice is trying to send data to Bob and they want to encrypt their data and they don't want anybody else to know what the encryption key is so they can decrease the data and the the spy it is traditionally referred to by the name Eve which is perfect because even as an eavesdropper and how do we do this well I've shown here the 2 possible states of polarization polarizes just a filter and you can't think of any measurement in quantum mechanics as some kind of filtration if I have a photon polarized this way up and down the
electric field is going like that and I have a polarized a at right angles to the polarizes cuts out that life and I did and if the polarizes is parallel then the light goes through completely unimpeded and I get 100 per cent I've got 1 photon I get the 1 photon and if I don't have 1 if I have the polarizes the other way I get 0 photons and that's perfect for computing because I can have this 1 and I can have this b 0 the trick here is not to just use that but at random 2 pick instead of what I call the Plus at the time spaces the Times basis is just the plus faces rotated by 45 degrees now if the polarizes this way I get and if it's the other way and the photons this way I get a 0 so I get 1 and 0 again but I have 45 degree rotation between this basis and this now what happens if I Have a photon this way and I have my polarizes this way while the quantum mechanics said at random I can't produce I get 50 50 1 and 0 and after I measure the photon is now instead of polarized this way this polarized you that wh or that way and that's the basis than of a completely unbeatable strategy to establish a key between 2 anonymous people are 2 anonymous computers connected up and then use that key to transmit data here's how here's what we do that I Alice since a 0 or 1 and at random and she also chooses the bases at random How could you choose the basis at random while 1 way would be for example to have a very very weak radioactive source nearby that's completely random if it's an odd number of counts in a certain number of intervals but did detector debts you pick the plus spaces and if it's an even number of counts you pick the time spaces the picked the bases at random but you recorded where you are what basis you're picking but you do not send that information and so nobody else not Bob knows what basis has been and then you send photons polarized that way and you can change that very quickly with an electric field and you send photons along to buy the baht has no idea because you don't want to to be telling you what the basis says that defeats the purpose of secure communication but there is absolutely no idea at all what basis Alice's pair therefore Bob doesn't try to figure out what basis analysis .period Bob at random he depicts plus or time as his basis and then measures the photon comes through with the polarizes reader oriented this way or that way and then measures iii the 1 or 0 but he has no idea whether what he's measuring with absolute certainty if his polarizes the same as analysis the theory of quantum mechanics says if I put a photon this way and I measured this way again nothing's intervened that I get the same value with certainty or it could be this way I'm getting a 1 at random and I have no idea we send a whole bunch of photon lots and lots and we can do that very very quickly so that and very cheaply you can have something about the size of this remote as with the laser and that no problem at all if Bonds basis is "quotation mark wrong in other words it doesn't match the bases that Alice picked at random then Bob just gets 1 a 0 at random but the problem is he doesn't know that he just gives 1 a 0 and how was Cedeno what on earth is getting the DOE has no idea what he's getting but what he does know is he's got a list of what basis he picked at random whether he picked plus at times and that's enough now bottles and send back equity measures the measure 1 a 0 in sequence but what he does send back 2 Is he sends back in a 1 or a 0 depending on whether he picked plus at times as the basis so he doesn't say what he measured it is on trial 1 I haven't picked plus on travel to I picked times on drought 3 it was times for was times by was class and so on and I have lots of them and balance of course has a list of what basis she but that information hasn't been sent and Bob hasn't sent what these measures and therefore Alice looks at Bob's choice of bases and whenever his choice of bases matches what she happened OK let's say with a certain trial and they match maybe many of whom don't match because it is random after all but if they have the manage them she says OK go ahead and pick these values so for example she records the the list and where they matched she then sends him another message which has nothing to do with anything that anybody can can use answers on why don't you use whatever you measured on the 7 the 13th in the 20 2nd and so forth trial photon and those are aware the basis that the but and but nobody knows what the measured result was because Bob never said what he measures but Alice knows what he measures because she knows that when the basis matched he got the same value that she got if nobody's eavesdropping therefore what they do it is they establish a sequence of ones and zeros that nobody else can know about and that's perfect to encrypt data they both know what the key areas they take normal data they encrypted with some scheme with these random sequences that only they know about and then they decrepit because each of them has the key but nobody else can help keep the how can they find out if something's going on well they can establish a key but they can have sent many many many more photons where the Basie's happened match were Alice was Plus and Bob was plots and what they can do after they've established the key is they can go back and say Hey you know tell me what you measured on trial a thousand 1 thousand 3 10 thousand and so on and if not if those don't match or if they don't match 99 . 9 9 per cent of the time then somebody might be listening so you can have a threshold that if somebody is listening with some kind of polarizes trying to trying to find out whether it's 1 or a 0 you can say Look up it seems that this line of communication is not secure maybe we've got an error in the detection system for the photon or something is wrong but we can't use this to talk about these sensitive things were going to discuss what we have to start over this scram whole thing and start over again and then go from there .period once you have the encryption key then you just use any old encryption scheme and the that key is that since nobody knew what the key was except Bob analysis by the clever way they didn't nobody can find out they can use a new key every single time may talk this is not like the Panel on an ATM card or a magnetic swipe that everybody can steal for every single financial transaction you do you walk up establish a new key and then in crypt all the financial data with that he that only you and the other side of the party the bank know about and nobody else can know about it and every transaction has its own key and so even if somebody tries to steal the data it's not like they can get on with with a single password and then look at all your stuff
because it's just completely hopeless and this is a cost power of computers is to do stuff like this to keep things tricky for somebody trying to steal your stuff cost the other side is that computers can be used to steal stuff pretty effectively and so anyway this has been used to secure method and it is being put into practice and maybe 1 day and every time we walk up to 180 on the Korolev a little laser will establish this key and then will take some money out and it'll be recorded OK so that's all I want to say about quantum crypto cryptography but it is an interesting subject simply because it really illustrates the principles of quantum mechanics and have your clever now took from 1926 until 1984 for somebody to figure out that you could do that but now it's it's a very old weighted to ensure How does away function involving time that's the question and the 1st point you have to make is that there is no operator for a time which seems kind of funny at least it did to me when I was a student because after all we can measure time or we think we can we can measure differences in time elapsed time it seems like something we ought to be able to measure but there is no operator four-time it's not as if the for example it's like position momentum there is no operator for time and there is no expectation value for time and therefore time in quantum mechanics is as we're going to treat it anyway it is just a number it's just a running variable like axes in position for classical mechanics it doesn't get elevated to us to a higher level and you might say well why is that in the short answer is that is time did have an operator than it would be her mediation and so that if you introduce an operator is not automation that you have a lot of problems well motion if that's the easiest way to tell of times going by as if sometimes if we see a car rolling we know times elapsed in the season the diving off the diving board into a pool we know that some time has elapsed and motion is related to energy an energy it is multiplied by time has the same units as H bar which has the units of action or jewels time 2nd and that dimensional analysis gives us a clue because when we had momentum and position we took momentum times positions it had the same units as age and now we've got time an energy and we take time kind energy and we get the same units a bars and so that gives us a clue as to what kind of way function we might want to to try to put in to start doing time-dependent phenomena well the 1 the kinetic energy of a particle is 1 have in squared or P squared over to well because remember for classical particle does not relativistic Pete the momentum is just and the east the square over 2 m is the same as 1 and the square and in elementary courses we always use kg for kinetic energy but I'm more advanced courses which is used to that's our notation and the potential energy which an elementary courses we call the potential energy we use the the potential energy of particle only depends on its position and the kinetic energy only depends on its momentum the 2 of them are totally different forms of energy of course they can be entered converted and we do that all the time but potential energy might be like a mass that is at a position and then if we drop it with synergies conserved we can figure out the velocity of the particle when it hits the ground when all the energy has been converted into kinetic energy the total energy that is equals T plus the and to cast this in terms of operators all we have to do is take our variables and we dress them up with hat and therefore we have be the energy total energy will be P had squared off over 2 m m is again just a variable it's not an operator just the P and then V well that's and the effects could be any functional form including just 0 but instead of just debts we put X hat and then we look at this thing and we say energies conserved over time if we have an isolated system and therefore this thing is going to stay the same it could injure convert between 1 former another but it can't disappear we can't get energy from below where it we can have energy go nowhere into nowhere is that could ever happen we know about it instantly
because we would have something that just sat there and ran and boiled water endlessly and didn't need to be plugged into the wall and that there would be very handy but unfortunately it's very impossible as well if we put in the explicit forms then of these operators the operator eggs have when it operates on the wave function just returns the value back and therefore when we can operate without sat on the wave function and we just get the of X where X is now a variable it's been turned into a variable there's no operator left P on the other hand was minus ii age times the derivative with respect to ax and I've got Peace Square and peace squared his peak times P like put in 2 of them and therefore peace squared over 2 m becomes minus age body by DX times minus side body by DX times won over 2 m times side-effects and that whole thing should then equal to be the energy times sigh of and that should be an equation that says that energy it is conserved and we now have to find the way functions that makes the energy concern and that will depend on the potential a we can tidy this up remembering that I squared is minus-1 and get my message bar square over 2 and the 2nd derivative with respect to acts of the wave function plus the potential energy of accidents the wave function is equal to be some number with units of energy times the way function and this equation is called the one-dimensional time independent Schrödinger equation this is an equation that says energy is conserved if you find the way function that makes this thing true you will have found be allowed energy but there is no time in this the equation yet now it depending on the functional form of the vexed the potential for example from molecular sprained what we might have the excess one-half kayaks square with so that the energy goes up quite radically that's the harmonic oscillator .period depending on what this form of this potential energy is electrostatic energy various kinds of repulsive forces and so on we can put all those in an if if we have something that we get certain way functions which if the particle is confined but if the potential gets big and the particles stocks like water in a cup and it has to stay there on electron-ion anatomy and what we find is that when we solve this equation we can't have any old energy and the reason why we can't have any old energy is that the wave function has to fit we saw that on the particle longer range and it's going to be the same no matter how it's trapped the wave function has to fit into the allowed space and that means it can only have a certain kind of wavelength not just any old thing and that means the energies at discrete entities get labeled with the quantum numbers so we have instead of the ESA Alabama and and we label them with the quantum number NO we don't have any time the question is how do we introduced time and that it's very tricky to think how to introduce time because we don't have any guidance necessarily from classical mechanics about how to do it and in the case of an isolated Adams let's say just sitting there in a vacuum in its lowest energy state it appears to just sit there "quotation mark and "quotation mark and the electron distribution the probability distribution of the wave function remains constant it doesn't fluctuate and that means that size stop side remains constant at all times and that's a big constraint them on what the wave function can do over time because that means if we use our capital wave function the dressed-up 1 with an explicit function of time and the size star sigh at some time t at at all values about has to be the same as side's sigh at some time where we call 0 where we start looking at the start of the experiment and that that means that there is whatever happens in time to some isolated state like that but it can't be too violent because if it were some strange thing that affected the way functional loss moved around a lot made extra lumps and stuff what would happen is we notice it we'd see some because with the probability of finding the particle and so forth would be different if and when we do the measurement of course we've we've destroyed whatever probability distribution was there but we can do the management over and over and over we can find lot the chance of finding electron is like a and it doesn't change if I wait a minute later it's the same and therefore that means that the most we can do this way function size is multiplied by a face factor e to the I'd say that why the I think well the length of the the fado is 1 it's just in the
complex plane an arrow with length 1 if they had a 0 it is 1 if they that is 90 degrees it's all right which still has the same like the 1 minus 1 and minus sign and so on and when I take sides start instead of Eagle miners I think I get you the plant site an ETA the anything times Seidel minus anything is either 0 and that's 1 and that means that the probability distribution stays put so now I have a clue when when I have a state with constant energy like this that's just sitting there then it must be that's all I'm doing in time is multiplying by this thing that has same length and so what I can imagine I have a distribution and the distribution could be moving in time somehow but rather than thinking of it moving like Jell-O and moving around what I should do is just color and it starts out quiet and then it turns gray and then it turns black or maybe it is starts red and goes through the rainbow Over time but its shape doesn't change and the only thing we can measure is not the color but just the shape following make the measurement no How could we get a face factor EDI well we know that anything in an exponential can have any units and we think it should depend on time so we could put in the exponential even the minus ii time something let's call it Epsilon and then because we we got Peter sex let's try epsilon time and epsilon must have the units of inverse time but which is not but even tho the probability distribution is stationary and we'd expect the face to depend on energy in other words although this thing staying put if the energy is high the colors are really flashing like crazy and as the energy is low the colors are hardly moving at all very slow throbbing and taking a cue from the presence of age Boren the momentum likened functions we could get us the following the city capital sigh at time t this capital sigh at time 0 times the to the miners I E. t over age by and that would be a very good guess as it turns out and because if we put this gas into the shrouding equations this weekend make a connection than with the time derivatives so now we take each side which was the time Independence Party which was just the energy and we say Well if the wave function is time-dependent will right there is b sigh at 0 times the minus ii the age by but if that's true than the other way of writing that to get rid of the E is to write I H bar just like we did with momentum only was minus side by Aidid's Pa time derivative of because when I take the time derivative outcomes miners ITT over bar without the key of course just the constant and so the age foreign minus over each part cancel and I get the the energy and that's that's exactly what I want have for using this on the right-hand side of the equation now we have a proper time-dependent equation which is very much like S E equals M the acceleration is the 2nd derivative with respect to to time of positions and what we have now is we have the kinetic energy the way function minors aged or square over 2 other times a 2nd derivative but outside plus the potential energy which assessed the potential energy times and that should equal in size but it is time-dependent then it becomes I H bar and time during the day of side and this is in fact the one-dimensional time dependent Schrödinger equation which is the basis for all kinds of time-dependent calculations that people carry out b wave function In this case depends on both acts and time and so to be mathematically rigorous we have to use the funny piece of writing the capital D this is not proper because he could have X and then I'd have to figure out what DTD axes and TEXT tea and so forth and that's not what I mean in this were taking the derivative with respect to space only on 1 side and on the other side were taking the derivative with respect to time only and therefore we have to use the funny duties and that means that this becomes a partial differential equations which can be very very very difficult to solve these these guys are bears and you take a course in PTE and you learn how to solve them when they're 3 spatial dimensions rather than just 1 then the kinetic energy adds up separately PX squared PY squared square no big deal and the potential then becomes a function of x y and z which we usually writers just are some factor which tells you where you are and then we get a fearsome-looking equation because we get minus 8 per squared over 2 and then this upside-down triangle the Dell squared operator on the way functions plus a potential and then we have the time derivative both outside on the other side as the same thing because there's still only 1 time to mention the Dalai operator this triangle thing is is just a shorthand because we we get so sick of writing by IDX where D by the y squared and then get carpal tunnel and give up religious right this triangle with 3 sizes the 3 derivative and the 2nd derivative of Adele squared is just .period Del the dealt with an arrow over it or boldface Dell is a vector it takes a derivative with respect to a certain position so for example if you're on a mountain it might be that if you walk this way but it's the slope is 0 it's the profits very the very slight and if you go this way the the slope is very fast and you fall off a cliff and likewise the derivative when you have a multidimensional function can depend on which way you're looking like extra Y and so you have to do that when you square and operator all you do is multiplied with it again and that means you operate with it again just as if you multiply by 2 twice you've operated by 2 squared or for now I suppose we have a free particle that means free particle there's no potential energy it's all and for example a neutron In a nuclear reactor which is uncharged and has no electric forces it's to neutral particle may be considered until it hit something like a moderator to be a free particles it's going it's very tiny going through it's mostly vacuum remember Adams a mostly empty space if you don't have any electric charges you don't notice any including you go right through which is why I read 1 reason why it's kind of tough to shield neutrons in some cases In this case I've put an explicitly here the kinetic energies the same age where over 2 hours the potential energy 0 so I put a big 0 and that's equal to the side at the and given the the kinetic energy of the neutron we want to find out the wave function side of the neutron but this is the 2nd order differential equation and may not be easy to solve if you haven't seen them before but we know what the kinetic energy of the neutron should be it should be the square over to emphasizes not relativistic and we know that the derivative of an exponential function any number of times gives back an exponential function and we know when we operate twice With the derivatives we get the wave function back times the number so we try I a solution like of it is the I acts upon age by and if we substitute our solution for this solution of the differential equation in than what we find is that it appears to work so I've just worked it out here for you the 1st derivative of that trial wave function gives a factor of IP over age bar it's the same way function back the 2nd rooted gives it twice and so we get minus P-Square squared over age bar square and of course the canal kinetic energy operator Excuse me has age bar squared over 2 and Sofia P. square age and we get peace over to went so that happens to work and so we find a solution that works at the square over to have the energy the way function is this evening I PX by and of course if we only expect the particle to have genetic energy and that's what we found the total energies P-Square square over to us what we didn't find is that there could have been a minus sign the exponent so I think the the plus side overreach but it turns out that I could pick the minus side PX by well that's just the thing moving the other direction the weird thing now is that the general solution is some a let's say times EDI Overridge plus another part B times even minus side accelerates bonafide put those in it and what parts of particle moving to the right with momentum the others moving to the left with linear momentum p and is perfectly acceptable 4 the wave function to be moving both ways at once and you might say will surely the particle can possibly be moving both ways at once what is the meaning and the answer is it means were doing quantum mechanics because a particle actually can until you measure it it's up to to itself what it wants to decide to do and so the same way that the electron can be slipping through both slid the particle can be moving both ways In fact that's 1 of the interesting things about these kinds of systems In the next lecture than what I wanna talk about this some one-dimensional model problems with certain well defined potential and use these model problems to show you with these equations that we've built up exactly why it is that atoms and molecules and various other systems quantum dots have quantized energy levels with discreet energies so we'll leave it there and pick it up electrified


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