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Lecture 02. Particles, Waves, the Uncertainty Principle and Postulates of QM
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welcome back to chemistry 131 a today were going to talk about particles waves the uncertainty principle and some of the postulates of quantum mechanics as you recall from the last lecture we found out at the turn of the last century by which I mean around 1900 that in fact the particles were behaving strangely and classical physics was not accounting for all the observations and so a new theory was put forward which came
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forward over a period of time called quantum mechanics and like any other fundamental
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theory it has to be compared with experiment and so there experiments that were done we saw some more modern experiments last time in which an electron in an electron microscope behaved very much like a wave the experiments ,comma mirror attaching and in fact this wave behavior had been observed and caused Luke Lewis too broadly and I believe it's actually pronounced embroidered but just 2 keep it clear will say broadly anyway he proposed in 1924 In fact that all particles have an associated wavelength that is related to their momentum and that this wavelength just follows the same relations have still excuse me the same relationship as that for a photon we saw last time that the photon momentum was given by H. and broiling proposed that In fact there was a wave length and land that was related to age upon P 4 articles not just for a photon and in fact in 1927 3 years later Davison and Girma showed that an electron beam fired a nickel crystal showed a diffraction pattern and that's the way phenomena and furthermore they looked at what the wavelength of the electron electrons in the beam would have to be and the wavelength was very very close to the exact predictions that the broiling made the question is where is the particles in classical mechanics the center of mass of a particle has in principle at least an exact locations at all times In quantum mechanics it's not quite so clear in classical mechanics the particle follows a trajectory what trajectory is is it's an exact specifications of the position of the particle and the momentum of the particle or its velocity of the masters at all times and that's in fact how we do all kinds of calculations in classical mechanics whether I'm going to shoot shall inherit land somewhere or anything along those lines and that works extremely well for large objects but it fails for small objects because they show this strange wave behavior now the
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problem with something that's demonstrating a wavelike behavior is that we can't say for certain where a wave is because waves tender spread out over time and so were kind of cot in a little bit of little difficulty because if we can actually say where the center of mass of the particle is if it appears to be blurry we can't specify what exactly that means that we can't have a trajectory and the trajectory following Newton's laws is exactly how you calculate where things are going to end up and so now you've got to have a new method to calculate how things are going to behave as as if they're showing this wave like phenomena and so that was a big chore actually and took a lot of very smart people a very long time to work out in 1927 and Verna Heisenberg made this 1 of blurriness more formal in the famous Uncertainty Principle which states that no matter how you design an experiment it is impossible to measure the momentum and position of a particle with arbitrary accuracy there is a certain minimum amount of uncertainty that's left over a given in this famous expression Delta P Delta exits greater than or equal to age bar over to it's important emphasize that this has nothing to do with your experimental apparatus and having some sort of deficiency and this is just an idealized experiment done as best as you could possibly do in this universe and you still cannot simultaneously specify the 2 things at once In fact the quantity H over to II so often that we invented the shorthand notation called H bar age with a last or age cross was created for and you'll see that often in the formulas that we use because we get tired of writing to Pike so much of the uncertainty principle says that we just cannot simultaneously determined position and momentum to arbitrary accuracy no matter what we do even in an idealized experiment and that runs counter to our everyday experience where we seem to be able to watch a rolling marble and plaudits mass and figure out where it is I'm basically about as well as we want to and so there must be something in this thing that that makes a difference for small particles in the something is the exact size of each bar if a Chippawa bigger we would notice all these things happening with big objects but age bars so small
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that we don't notice it the same way we don't really notice the momentum of a photon otherwise the lighting that is on me now would be pushing me around and I'd have to fight like a mind to keep my position now the rationale is when we think of looking at
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something we're looking at something with the lights on we can't look at something in the dead pitchblack and see anything but if you've got a small particle then we've learned that light consists of packets both times and these have momentum and energy and so when we tried see where a very small particle is we can't just look at what we have to
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bounce something off we have to use something like light and the light itself is going to change the momentum of the particles and if we want to get the position to be very close then we have to use a short wavelength of light but that's a high frequency and we learned that the quantum energy of light is age New and so if new is high that means that we're going to have come in with a kind of energy like an Xray and then they're going to boot the particle around and so although we could say it just was there now we can't say very well what its momentum and these these ricochets happen all the time and so we have a fundamental problem if we try to do it now if we turn off all the lights then we know that the particle is moving with a certain speed but then of course we can't help at all where idiots so for small objects the photon kicks the particle and that creates the fundamental problem so it's a bit we don't really know how big an electron is it appears to be a point if you do expects periments with it but to tell where it is we need than a wavelength of light that small so just the same way as you put on infrared goggles and you look at nite but you can see heat but everything's much blurrier because it's not as sharp as visible light because it has a longer wavelength so that we need a small wavelength of light to get a particle down too what we would consider to be a reasonable precision of measurement and that gives a big kick to something like the electron marks and so the electron momentum in that case becomes very uncertain conversely can take over how you might have thought design an experiment that would measure the momentum of a particle 1 way to do it would be to have a very very long thin and have particles coming in and if they are going straight along the 2 then they get hit the wall and their they're out of there and you can have some shoppers like fan blades at certain distances and moving with certain rotational speed and if the particle happens to be going the right speed so it goes through the whole of this fans and then the whole that fan and so on then you know for sure that it has a momentum within a certain range so you've isolated it very well but if you really want to get it to be very very small uncertainty that means you're going to have to have a very very very long to and then the particle could be anywhere inside the 2 tube so you don't really know its position and of course if you open it up so it's not dark polite and then that follows up the momentum as we as we said before so we can specify the position of any kind of object without light To see it or something else which would be even worst problem now from macroscopic objects the uncertainty principle does not limit us were always limited experimentally for anything we do that sort of size we can see it has nothing to do with that but for very small objects like a single electron it becomes the major factor so here's a practice problem that's meant to illustrate this difference supposed weeks of practice problem 3 suppose we take a 1 grand and we know it's position to a 10th of a millimeter which is pretty good and we know the position of an electron to 200 Pico meters which is 200 times 10 to the minus 12 meters the question is what would be the minimum uncertainty according to the uncertainty principle for the velocity in each of these cases so 1st of all the uncertainty and momentum it is in the velocity not in the mass the mass stays fixed so Delta P becomes an adult acts and then we use the uncertainty principle Delta P Delta X is greater than or equal to age over for pie we have to be a little bit careful with the units we were given the units in GM but we have to convert 2 kilograms and we were given the uncertainty in mm but we have to convert 2 meters because we're using MKS units and if we take account of all those factors that we find the Delta V is H which is 6 . 6 2 times 10 to the minus 34 over for pie and then we have 10 to the minus 3 kilograms that's R 1 gram and tender the minus 4 meters and if we work out the unit's carefully we find that the uncertainty in velocity is 5 . 2 times 10 to the minus 28 8 meters per 2nd for reference and an Adam is about 10 to the minus 10 meters and so the uncertainty and velocity is way way way way way way way smaller than anything we could possibly ever notice it was just as good as almost infinite infinite precision as far as we're concerned for the electron however if we do the same calculation and hear the differences that we put in 9 . 1 times 10 to the minus 31 kilograms and we put in 200 times 10 to the minus 12 meters the what we find is that the uncertainty and velocity it is about 2 . 9 times 10 to the plus 5 meters per 2nd so that's 200 thousand meters per 2nd if you've ever run at 10 K you realize that that's running pretty fast and so trying to localize the electrons down to just 200 P commuters means that its velocity becomes very very uncertain so the uncertainty in velocity is very very large and and the reason why there is a large uncertainty for the case of the electron is that the electron has such small where but macroscopic objects in the 1 gram arranged personal property OK now we have had to broadly said Look matter has a wave associated with it but which is kind of a semantic dodge it's not as if matter has suddenly become a wave we still can detect particles in the Tottenham experiment when they hit the screen they given .period and we know how to characterize waste there's a face frequencies it's an amplitude and in fact wave equations were known from electromagnetic
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radiation Maxwell's equations which seemed to indicate the light was away even explained many many of its properties was a wave equation and so physicists knew how to write those things down but there's kind of a question here as to whether this associated wave is the real thing or not a real thing is that the calculation all device or is it a real thing while Davison and Girma seemed to indicate that it is a real thing that this too broadly wavelength and for small particles this can be the main thing depending on what kind of measurement you're making so the the the question is is that the broadly wavelength and actual measurable things and if it is then what equation do we write for the way and finally if the wave if the behavior is wavelike then how do we explain the sharp spots In the town of mere experiment when we wanted the electron to sift through both of this lets say it was convenient that to think of it like a wave because the wave can do that and a particle cannot but when it hits the detector it seems like it collapses like a poorly camping tent on to just a single point on the detector and it certainly doesn't light up the detector like a wavy I think we see the sharp spots and only after we measure a lot of these measurements do in fact see that 3 aggregate behavior is waiflike each individual 1 doesn't look quite like but this took a lot of thought to sort out for sure but we don't notice any wavelike behavior with macroscopic objects we don't notice any yellow we move around we don't notice that the waves coming off my hands that I can't tell where my fingers are or anything like that and the reason why is the same reason as with the marble that the small value of plaques constant is the explanation where just so much bigger than age far that we don't notice that and for an illustration was to another practice problem practice problem for a let's figure out that the broadly wavelength of 1st of 5 milligram grain of sand moving at . 1 centimeters per 2nd flowing along the beach and 2 electrons moving at 1 kilometer per 2nd well for the grain of sand we just plugged into the broadly wavelength formula landed is age upon the and then the rest of it is what a lot of chemistry it's it's units conversions keeping track of the units crossing them all out and making Donncha at the end that the units are what you intend to be so in this case I explicitly rolled out a jewel is a kilogram meters squared per 2nd squared cross out the jewels crosssell kg make sure the MG is turned into kilograms make sure the CM turned into meters and I find that the broadly wavelength for the grain of sand it is about 1 . 3 times 10 to the minus 25 meters that means the Associated wavelength is much much much much smaller then any nominal size we would think of of a grain of sand which is certainly much bigger than at however for the case of the electron but a broadly wavelength working out the same math that using the electron mass and using the velocity given now converting kilometers 2 leaders it works out to 7 . 3 times and minus 7 meter anonymous tender minus 10 meters and so now we're talking about a wavelength of the electron moving at the speed that's much much bigger than anything we would think of as the size of the electron because even if we don't know exactly what the size of the electron is we know that Adams have electrons in them and therefore the size of the electron has to be much smaller than the size of the ad where they'd be like beach balls and they wouldn't fit and so in this case when we have this situation when the wavelengths that we figure out from the broadly formula is in fact much bigger than any nominal sizes of whatever it is we're considering at that point we have to say watch out because whatever this thing is it could surely show "quotation mark behavior when the broadly wavelength is very much smaller than the size of the thing Our idea of the size of the thing then that we are going to see any kind of wave behavior and we might as well just cut to the chase and use classical mechanics to figure out what's going to happen so that's summarized in this next bullet .period we don't notice any wave behavior if the particle is has a very small too broadly wavelength and if it's larger than we we surely do notice that at least in some experiments we were now what are we going to use to actually figure out what something's doing for for particles we had Newton's laws we had ways of figuring out as equals and and so forth but was going to happen now we had this too broadly wave wavelength but we
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still don't have a way and we don't really want to call the particle wave because that's a little bit confusing it makes you wonder what a particle ever was and so what we do is we have kind of a little bit of a Dodge we speak of the wave function other particle and way function is associated with it as given the symbol side to describe its behavior and whenever a chemist want to keep our people they switched to Greek letters because of his wish to Greek letters that seems much much more intimidating and you have job security but that the sample size is universally interpreted as a probability amplitude and I'll get to that in a minute and the absolute square we swear we get a probability density and there is a reason why were using probability and not certainty and that part of that has to do with uncertainty and part of it has to do with the the very nature of measurement as will see in the end of this lecture possibly in the beginning next lecture we can only know quantum mechanics as the probability not the certainty even as frustrating as that might seem of finding the particle somewhere Due to its wave nature it appears to be spread out and when we measure it we interfere with that somehow and that causes the measurement to collapse like the tent I mentioned but the probability is uncertain until we finally make the measurement it's as if I had taken a die and I throw it and I'm not looking at it and then it lands in it bounces and rolls around and so forth and then I look at it and it's a 2 and I said Well the probability beforehand of getting a 2 was 1 6 but in fact now that I see it's to the probability is 100 per cent and we're going to see that phenomenon when we make a measurement once we know then all the other possibilities seem to have just vanished somehow which is a little bit mysterious but that appears to be what happens and that's a common interpretation anyway if the theory once we measure the position we can yield different values and even tho we I have exactly the same electrons coming through the 2 slips and everything is exactly the same each time and we believe all electrons are identical we still get a distribution of different values we don't necessarily get 1 died in fact we suffer sure we didn't get 1 don't we get dots all over the place
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and when we added them up it was much like looking at a billboard when you to close it's just stopped when you get back you see the big picture if you have a lot of and then you see that it looks like a wave and keep in mind that this is in a hypothetical perfect experiment it's of course not possible to do a perfect experiment but it's possible to beat to get very close to a perfect experiment with some kinds of setups if you're careful and even then you get a distribution of results which is just far far greater than any uncertainty in whatever you set up and so that looks like there's something else at play and you might think well maybe there's maybe there's no maybe there's some kind of thing but we
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don't proceed and it's there like road noise you only notice its absence when you go camping but you don't tend to notice it was there you know it's always there so maybe everywhere when we're shooting these electrons and photons there is some kind noise or something else around and if we were smarter and we figured out what that source was what was jiggling things around them we might be able to get rid of it and then we might not have this theory of quantum mechanics we would have a theory where we could measure things the way we want to and so forth and so on and in fact it seems like that's not the case that I'm quantum mechanics really says that this probability it is the nature of nature and not the result of an incomplete theory by leaving out something it but we just couldn't figure out I want a 1 of the biggest luminaries so far in physics didn't like this theory Einstein famously said God does not play dice with the universe keep in mind that that very bright people can be wrong about things Linus Pauling who is 1 of the brightest guy so far in chemistry and had a theory that vitamin C and huge doses would prevent prostate cancer and that that was you know he believed ardently and in this case apparently even Einstein was incorrect in fact it seems like there's nothing but probability there is nothing else for example radioactive decay doesn't depend on pressure temperature or anything else that's been explored to any appreciable degree that's why we can use it to date things and figure out how old something is and so forth and so on and we believe that all the nuclei are identical they all have the same number of protons and neutrons and there they are in the sample and yet all we can know Is the halflife which is the time for example it for half the sample atomic nuclei to decay away and to some other elements and for carbon14 the halflife is about 50 100 years which makes it convenient for lots of measurements of things that have been around since humans have been around but not too useful for something that might be extremely old because the national carbon 14 left and so when you try to counter you just see nothing and you can say it has to be older than this but you can't really narrow it down to much but if we luck in are identical sample and we watch them there is no way we can tell which particular nuclear Of all the identical said going to decay we can't beforehand make any kind of experiment that's going to tell us that all we can say is that in 50 100 years they're going to be half as many as were today and so this this means that really probability is the whole thing when it comes to something like that so there is some possibility of quantum mechanics we need to understand them because we need to have this new theory down pat we need to know what assumptions and it's making and these postulates formed the basis for all of our understanding and for all the detailed calculations that we might undertake the 1st parcel it is this at any time the state of a quorum system is described as fully as possible by the way function side which depends on the coordinates of all the particles that make up the the system these could be the electrons and the nucleus of an atom or something bigger like a molecule but in any case if we know side we know everything that it is possible to know about the system and what that means comment 1 since the wave function contains all that we can it follows that most of the time we do not know the way because in order to know the way function we would have to be making lots and lots of very clever measurements and
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usually we don't so usually we just know the dogs in the yard but we don't know exactly where it is so there will if we say we know the wave function of a system where making a very very strong assertion comment to were often interested in wave functions for quantum systems that are not changing times like for example the properties of an isolated Adam on isolated molecule and in that case we use a lowercase side to denote the wave function but in some cases we might be interested in a timedependent phenomena and then we use in upper case sigh and the problem is they look very similar I usually put but alike Ussery refund and on the top 2 2 indicate that you're using the upper case sigh and usually set time apart as a separate variables so time is an input to the wave function you have to know the time to calculate the wave function any as I've written on the bottom of slide 52 here the question is How can we suppose that the particles whatever they are have exact positions R 1 R 2 or 3 and so on when a couple slides ago which is decided but there is uncertainty in the position of any quantum particles this is mess kind of using it's sort of bad logic here to to be assuming this and the answer is no it's only with respect to measurement but we have to worry about the uncertainty principle and even then it's only for a joint measurement of something like position and momentum along the same coordinate axis exposition X momentum the variables here the xyz coordinates of all particles are better view just a simple parameters on which the wave function depends I once we've calculated the wave function then we can use it To desired all our measurements so we're gonna make it and magically everything comes out just hunkydory parcel at 2 makes an assertion about probability it says now what this way function me the probability of measuring the position of a quantum particle at some positions let's say X not within some small region In small enough that the wave function doesn't change value very much over the small regions is given by size so I stopped outside the DX or modulus sides squared DX for a 1 dimensional system for a threedimensional system we have denigrated overall the spatial variables TEXT why DEC on polar coordinates the RD favored the fight and in any case now we have sigh are not squared times DVD some books used detail I prefer TV because it reminds me that it's falling so our 1st comment is what it is the asterisk What is the size star the answer is that the asterisk denotes the complex conjugate because the wave function is often complex now at 1st blush that seems like that might be a problem because I'm you're saying that this thing that's associated with a particle it is got an imaginary part To win but in fact not really because I can write down a simple algebraic equations let's say X squared plus 1 is equal to 0 and that has a solution but it has an imaginary part so if I just say Well I don't wanna have any imaginary numbers at the end and in the wave function it might be like Algebra I won't have a theory of rooting polynomials or anything left over in my theory so we accept this but of course we realize probability is real and that's exactly why we take the complex conjugate so if you're given a complex numbers equals X X is called the real part plus I Y is called the imaginary part than the complex conjugate is obtained by just changing the sign of the imaginary part you can think of a complex number having an annexed part and a wide apart and it the
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complex conjugate the experts as saying that the is reflected to the other side so if it's up up
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here goes down of it's down here it goes up and you just do that wherever you see I mechanically and it's very straightforward to do so it's not a big deal in terms of calculation and if you then work out was Z's stars is or absolute C squared the modular square you'll find that it's X where plus Y square because you have to recall that I squared is equal to minus 1 and that's exactly what we want because then it's a real positive number and if it corresponds to the length of something and usually when you have the length of something that means you're going to be able Adam up because length that's and in our case is going to be probabilities that are going to have to add up and they're going to have to add up to 1 comment to the probability of finding the particle somewhere in the universe should be 100 per cent that is the particle shouldn't disappear and some cases particles are annihilated and may turn and other things but
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we are going to consider those cases in chemistry those for physics in terms of us if we have an electron the chance of finding it somewhere has to be 100 per cent at least in principle and that means that there is another constraint on the way functions the wave function should be normalized that is the integral from minus infinity infinity of sigh of X square the Act should be equal to 1 and that means that of course sigh of X whatever it is has to be a function that we can integrate and it'll turn out that it should also be a function that we can differentiate as well so we can figure out where things are going in time and so forth and that means that sigh of X is really in mathematical terms a very wellbehaved function is not any exotic mathematical function that would would cause causes problems Of course we put these limits on the integral plus and minus infinity and we do not think the universe is infinite or rather we think the universe is not infinite but the mass that we do is much easier when we make the limits insanity this is often true in all kinds of fields where if you have an infinite sheet of charge it's very easy to calculate the electric field and so forth and if it's a finite she it's much harder their terms to subtract and if there's a funny shape she it's really really hard and it doesn't teach you anything necessarily different and so just to get the principal town usually take a simple case and often it's Infiniti so we're going to assume that the universe is infinite that won't make any difference for our calculations parcel at 3 for every observable property that we can measure energy linear momentum position angular momentum there is a new player in the game it's a linear her mediation operator that acts on the way functions but now this is a new object that many of you may not be acquainted with so I want to take a little bit of time and then explain what's going on on operator is like the Big brother of a function when I think of a function I think the function grabs an input and then returns a function value so it grants a number and it gives back a number and operates so like equals that of experts an operator takes in a the whole thing and then gives a new function and there are things that take in operators and you give new operators and seeking keep going but the effort in terms of this course
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we don't need those other objects so I've written here Omega With a takes in the function f of x and returns the function g effects In this course operators are going to be like gentlemen in the 1950's they're not going to show up without a hat and usually we just omit the extra set of parentheses we keep the hat on so we know we're talking about an operator and not just a number something else and we simply write all may go to the last Of the function always to the left because it's acting on it to the right all major acting on give us chance and operators can be as simple as just multiplying by function by X or even a constant because that is a new function or multiplied by 1 so we get the same function back still UN operations so here I've written X hat the operator banks had operating on death give the variable X
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without the hat times enough and that's a new function G so if its efforts X than X hat on after his x squared and so on once the operators done operating the result is a new function which just has variables which might be the same so 1 of the things you're supposed to do and quantum mechanics when you see an equation with an operator in it is you're supposed to let the operator do its work and then get back to just functions that you can differentiate and integrated and so forth that's that's the goal don't leave the operator hanging around unless you have to comments for operators have units multiplying by Iraq's it's going to add length units to the new functions in a chemistry we have to be careful about units we can't just be multiplying by things and not know what the units are we have to make sure we get the right units whether it's energy year momentum position this leads to practice problem 5 which is the following does the wave function have unit and if so what are they well let's go back to what the wave function was we know the integral the wave function squared represents a probability probability is a ratio and has no units we wrote for a normalized wave functions in the goal stock side is equal to 1 but the integral was against Andy axes like X it has units of land and therefore sigh stars side whatever it is musthave units of inverse length or a link to the minus 1 and therefore sensors 2 of those guys and changing the sign of the imaginary part doesn't change the units sigh itself must have units of length to the minus onehalf power or 1 over the square root of flank for a threedimensional way function so I had to have unit of length to the minus3 have power the expectation value or average value of any observable once you know the way function is given by an integral of a sandwich with size
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star the operator and sigh on DX This is for onedimensional problem the expectation value is usually denoted with brackets around the thing which means an average value or the value we expect With a very very large number of measurements but no single measurement need ever returned the expectation value
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for example if I slip an unbiased calling and I count 1 every time it comes up here and I can't 0 every time comes up tale then the expectation value of I make a very large number of passes is 1 but we never get 1 onehalf in any of the measurements we do we the dead 0 or here's a challenge if you're interested a little bit harder what's the expectation value for throwing a pair of dice a large number of times if you can do that kind of problems you may have a history In playing craps the other gambling game parcel at 4 the only possible result of a perfect measurement it is 1 of the eigenvalues of the operator corresponding to the measured observable that's quite a mouthful what does that mean well I Eigen functions and I can values the central mathematical objects in the theory and that's 1 reason why you want to take math courses up to and including linear algebra so that you can learn about these things without having to learn them on the fly while you're also trying to learn something else about the subject and I can function is a function which the operator returns unchanged except for multiplication by a number With units 1st of all some form of the Eigen value equation I've written here Omega had on a give little Omega which is a number with units on but the main thing is but is the same function and all major is a constant Colby died in value and what we want to do in quantum mechanics often it is given an operator we want to know but it's Oregon functions are because the result of a measurement is always 1 of the eigenvalues of the operator and if we don't know if I can functions we can calculate its Oregon values very easily so due to go back to the diet I suppose we don't know anything about it we might think what we could get 1 and a half for an answer because we don't know how it shaped we don't know what it is but in fact therefore if we look at it it's got numbers integers 1 2 3 4 5 6 those are the only values you can get by tossing a onto a table you can get something else and just knowing that those are the only 6 possibilities that you can have is knowing that time compared the not knowing anything they're thinking they can be whatever values you might dream up so given an operator what we have to do we often had the task of finding all the possible functions and all the possible values all major don't make the Eigen value equation true In mathematics the satellite and values is called the Eigen value spectrum has just for you aficionados but still practice problems let's consider the derivative operator d by idea what's the set of Icahn functions and I can values for this operator well we set the operator to the left of the function the functions unknown in the iron values so we're just going to say the derivative of effort is equal to the times after and this site value equation than in this case amounts to solving a firstorder differential equations and my advice is that another very good math course to take so that you know how to do it In this case it's fairly easy to do we separated variables and we write upon us equal to CDs and then we put in a girls on both sides and that we realize that sees a constant the Eigen value that does not depend on access and therefore we can move C on the outside of the integral we can look up the entire derivatives use mathematical if he's had 5 to do the angles or you can actually put immigration like that and integral Goldstone ,comma online and it'll solve it for you and Willie you'll find that the natural log is equal to the CTX plus some constant because this is an indefinite integral and if we exponentiation both sides we find the function f of x is equal to the to the CX plus we can factor that out this year's EX time to see and then we can let the to the CB some constant K you did the CTX and as a check we can always substitute our solution into the original equation and see that it satisfies in fact what I was doing differential equations 1 of the most powerful methods was guessing you guess the solution put it back in and see if it works out and I can often be quicker than to try to do it the the forward wh so here let's do this the derivative of mathematics Is the derivative of K E to the year and that's Cape tons ability the CX even I know how to do the derivatives into the EX some of the CX rather that's Kasey needed the access and that's just the kind that so we've shown that the operator operating on give a number C that doesn't depend on current effects that's exactly what we have to so the I function of the derivative operator is the exponential function and the item value can be a complex numbers as long as it's a constant and the reason why when you solve differential equations but the exponential function occurs every quick In the solutions of his equations is because it's an item functions Of the derivative operator but 1 commenter derivative operator is linear the integration operators linear and by linear I mean that's the derivative of alpha acid "quotation mark stated Jews Alpha tons of derivative of death plus beta times a derivative of G Freddie functions a F and G and any Constance Alpha and data however the derivative operator is not her mediation and to go back to how we're going to characterize observables the idea was we had to have a land here Her mediation operator now we know what linear means we have to figure out what her musician means and has to satisfy the following relationship the integral the best Omega had G is equal to the complex conjugate of the integral of GStar Olmert had so we put a stop and we swapped the order of the functions if it follows that it's her mission and here that engineer any reasonable functions that inoperable in the usual sense OK that's quite a bit for today so we're going to take a break and come back tomorrow and we're going to pick it up on what her mediation operators are cover a couple more parcels in quantum mechanics and then do a few interesting problem
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Metadaten
Formale Metadaten
Titel  Lecture 02. Particles, Waves, the Uncertainty Principle and Postulates of QM 
Alternativer Titel  Lecture 02. Quantum Principles: Particles, Waves, the Uncertainty Principle and Postulates of QM 
Serientitel  Chemistry 131A: Quantum Principles 
Teil  02 
Anzahl der Teile  28 
Autor 
Shaka, Athan J.

Lizenz 
CCNamensnennung  Weitergabe unter gleichen Bedingungen 4.0 International: 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 und das Werk bzw. diesen Inhalt auch in veränderter Form nur unter den Bedingungen dieser Lizenz weitergeben. 
DOI  10.5446/18880 
Herausgeber  University of California Irvine (UCI) 
Erscheinungsjahr  2014 
Sprache  Englisch 
Inhaltliche Metadaten
Fachgebiet  Chemie 
Abstract  UCI Chem 131A Quantum Principles (Winter 2014) Instructor: A.J. Shaka, Ph.D Description: This course provides an introduction to quantum mechanics and principles of quantum chemistry with applications to nuclear motions and the electronic structure of the hydrogen atom. It also examines the Schrödinger equation and study how it describes the behavior of very light particles, the quantum description of rotating and vibrating molecules is compared to the classical description, and the quantum description of the electronic structure of atoms is studied. Index of Topics: 0:00:20 Louis de Broglie 0:02:32 Where is the Particle? 0:14:49 Waves 0:19:20 Practice Problem: de Broglie Wavelength 0:21:16 Wavefunctions 0:29:16 The Postulates of QM 