Lecture 03. More Postulates, Superposition, Operators and Measurement

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Lecture 03. More Postulates, Superposition, Operators and Measurement
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Lecture 03. Quantum Principles: More Postulates, Superposition, Operators and Measurement
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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: 0:00:19 The Postulates of QM 0:05:30 The Momentum Operator 0:08:17 Basic Functions 0:13:44 Orthogonality 0:28:50 Uncertainty 0:30:51 Complementarity 0:35:00 Classical Atoms 0:39:12 Wavefunctions and Orbitals 0:43:37 Confined Systems 0:45:49 The Position Operator
Sense District Reaction mechanism Clenbuterol Ageing Magnetometer Orlistat Gemstone Active site Organic food Derivative (chemistry) Process (computing) Chemical formula Area Potenz <Homöopathie> Sense District Operon Arginine Cubane Ice front Stockfish Wine tasting descriptors Vancomycin Functional group Cardiac arrest Derivative (chemistry) Calculus (medicine)
Elastizitätsmodul Combine harvester Operon Diamantähnlicher Kohlenstoff Functional group Optische Analyse Diamantähnlicher Kohlenstoff Shuttle-Vektor Nanoparticle Multiprotein complex
Left-wing politics Plant breeding Wursthülle Man page Meat analogue Magnetometer Enol Shuttle-Vektor Conjugated system Hexamethyldisiloxane Stop codon Operon Arginine Walking Biofouling Adamantane Ice front Multiprotein complex River source Stockfish Tanning Wine tasting descriptors Electronic cigarette Lava Combine harvester Gesundheitsstörung Functional group Optische Analyse Azo coupling Blind experiment Boyle-Mariotte-Gesetz
Systemic therapy Stop codon Magnetometer Functional group Lactitol Chemical property Zunderbeständigkeit Boyle-Mariotte-Gesetz Mixture Tanning Man page
Ionenbindung Set (abstract data type) Wursthülle Radioactive decay Atomic orbital Mast cell Hydrogen Molecule Meat analogue Electron Magnetometer Enol Match Kernproteine Faserplatte Cycloalkane Periodate Surface science Pitting corrosion Rock (geology) Reaktionsgleichung Chemical property Walking Legieren Adamantane Protonation River source Herzfrequenzvariabilität Man page CHARGE syndrome Knot Wine tasting descriptors Meat analogue Systemic therapy Gesundheitsstörung Atomic orbital Match Functional group Optische Analyse Atomic orbital Proton-pump inhibitor Wasserwelle <Haarbehandlung> Atom Calculus (medicine)
Sunscreen Set (abstract data type) Wursthülle Atomic number Man page Molecule Sample (material) Island Electron Magnetometer Sand Singer Corporation Match Cycloalkane Derivative (chemistry) Nanoparticle Mineral Cork taint Area Chemical element Operon Pitting corrosion Chemical property Chemist Wine tasting descriptors Wine bottle Systemic therapy Emission spectrum Gesundheitsstörung Spectroscopy Atomic orbital Functional group Valence (chemistry) Common land Wasserwelle <Haarbehandlung> Thermoforming Atom
welcome back to lecture 3 can 131 and they were going to continue where we left off today it's more parcel superposition operators and measurement where we last left our here we had decided that the derivative operators Lanier but it was not her mediation and then I introduced this very ornate and relationships to describe what I meant by her mediation and you might wonder what it means but what it means is basically suppose we had complex numbers and most numbers were complex and then we wanted to that number was but we only a complex numbers while 1 trick we could use as we could say if is equal to the star then the number is real because the only imaginary part that can be equal to opposite itself is 0 and 0 imaginary part means numbers real and so really her mediation is just making sure that when we measure something we get a real number we still do you believe the probability doesn't have an imaginary part and neither does energy but when we measure it has units jewels and so forth and so we want to make sure that these things that we measure our mediation and this formula with Senegal's stars and the operator and is just a very fancy way of saying sees equal to stock are nothing more than that OK let's show that the derivative operator then is not her here's what we have to do we have to do an integral part of a star d by the X G and we have to show that that is a is not equal to the inability G-Star D by D X yes the whole thing star when you see into grand but has a derivative in the 1st thing you think it is I bet I can integrate them by parts if you recall integrating by parts is basically doing the opposite as doing the derivative of UV is UTV plus BTU we turned that around and we moved the UB no 1 of them to the other side and then we set the integral equal to that now the limits on this integration a plus and minus infinity but I will always put them in but because it may get a little bit messy but whatever it is the wave functions have to venture of plus or minus infinity and the arguments as to why they have to banish is if they had any amplitude out there way out there then we could normalize them they would get too big and so the only way we can have the area under the curve cranked down to some number is that it finally dies out when we get far enough away the let's try integration by parts of the formula is the inability you the by the acts of the effects is equal to use -minus there go the other way around these dividing X you and this is going to be convenient because the Hermitian then had them the other way around analysts let you the function you of acts conventionally in calculus led the lesson that B stock and the let's let that be BG and let's try so our equation becomes this fairly intimidating looking but not too bad the illegal arrest started the by DX G it is equal to S. stodgy evaluated plus or minus infinity -minus the integral G the bottlenecks at Star and that is equal to the integral of G-Star the whole thing start but that's not equal to what we want and so what we have because we have a minus sign and we wanted to be a plus sign so it's not equal and therefore the derivative operator is not her Misha it's called isometric for obvious reasons when you swap on the changes sign but it's not her mediation How can we have a bigger mediation while interestingly enough we have to use our friend the square of negative 1 again and if we multiply the derivative operator by minor side each bar that's enough to do the job because minus star this plus and so therefore that gets rid of the minus sign that we got stuck with With the regular derivative and then we just follow everything else you and I works you say Well what lies aged father and the answer is this is quite a mechanics a courses age body because we're going to have to have that in almost everything we use and in fact the momentum operator P hat acts which tells us the momentum when operates on a wave function tells us the momentum in the direction is just given by -minus H bar the by DX and it is a linear mission operators it cited functions are very closely related to those of the derivative operator because after all it has it's just an extra thing up front but we want to make sure that the Eigen values are real and so the site and functions are exponential sp but we put in an ally and here we realize that P is real X is real a spot is real and so this is EDI PX upon each bar we know that we have to have the units go away if we take an exponential because an exponential is a power series 1 plus an explosive were too and if it has units were adding feet and feet squared dance the Cuban that doesn't make any sense and so With a little bit of dimensional analysis we come to the idea that these functions here either the I acts upon each bar are very good candidates so let's do another practice problem and have a look so let's show that these are the Eigen functions in fact of the derivative operator well let's take he had acts on our function FedEx but put on what he had in mind inside body by DX on the function was put in the function which we assume is either the plus I PX upon bark and the derivative of anything times axes the thing excuse me anything top-seeded a axis a DAX so we bring down the IPE upon and I think you can see why we want age from the age bars fold up -minus NYTimes plus is minus squared but minus ii squares plus 1 that goes away and that leaves us with the and that's P E to the IPO acts upon its bar recipe times the Eigen function and therefore we have shown that the operator P had returns the Eigen value P which has the units of momentum so the complex exponential is the organ function of the momentum operator and the item value this peak In the language of linear algebra the Eigen functions of a linear Hermitian operator former basis so if I take a point in a two-dimensional plane and I want to figure out where I am I know that if I go a certain unit out on that along the x-axis and an
upper down by what that I can get to the point and furthermore at any point in the square can be expressed as a combination of some distance this way and some distance down and there's no point that can escape before we have a vector any director any point X not why not that's equal to X's not times the Coronet unit along the x-axis plus why not tons according unit
along the Y axis and just like that we can write anyway functions as a linear combination of Eigen functions over her mission operator they form a basis no function can escape and that's important because if some functions could escape that would mean there were certain the values that we could measure anything and I would be very bad because with what would happen to the probability particles would be disappearing well past 2 at 5 is this is quite a mouthful but will get to it one-way wave function is not a I function Of the measured observable the result of the measurement is still alive in value but now the probability it is given by the square modulus of the expansion coefficient Of the Eigen functions as the operator so if I have always functions size is some constant could be complex number doesn't matter because all these functions can be complex Celestina Scott C 1 3 1 plus C 2 feet too then the probability of obtaining the 1st item value is the square of C 1 With the with the absolute value so there's a measure party takes the one-star and probability of obtaining the 2nd 1 is the square of the 2 those are the 2 probabilities and if there are only 2 parts making up the wave function and those are the only 2 values you can get usually away function is made up of a whole bunch of different I can functions and so there are a lot of different possibilities that you can the item functions themselves have to be normalized that means if you happen to be in enlightened functions your chances of being somewhere in the universe and having that item valueless sales momentum is equal to 1 and so this the basis functions themselves are normalized and we always assume that they are normalized without comment and likewise for the wave function to be normalized once the basis functions are normalized that means that the probabilities in those coefficients to add up so the sum of the squares of all the coefficients always have to add up to 1 it's as if we have a unit circle there were some point on the circle and we have an X component and a Y component than the Pythagorean theorem says X squared plus y squared is equal to 1 and that's how it works and it works the same way and higher dimensions 2nd ,comma the best way to think of these Oregon functions is not as things spraying around in space think of them as vectors think of 1 I can function pointing this way telling you the amount on this side the other 1 points this way a 3rd 1 point His I've got
more I have to have an imagination but basically they're all at right angles to each other and they're all telling the amount of this special state that is in there To begin with and I can functions of a linear her Hermitian operator corresponding to different eigenvalues are orthogonal and that's another reason why it's good to think of them like the actors because of overnight in function here and this has 1 -dash value and I have another item function here is divided then those 2 functions have nothing to do with each other they are as different as different can be there in different directions they have no influence on each other and to see this normally suppose we have X and Y we can tell their right angles because we can look the suppose I put my arms out and then I say well are those orthogonal while you could try the mentally rotate and see if it comes back that's wide but that's a very very slow and labor-intensive way to do it instead what you do is you take the dot product you take the product of the 1st 2 components the 2nd to the 30 Adam all up and you see if that's 0 and if it's 0 0 that means they're orthogonal if it's not 0 so that means that they are orthogonal so for 3 real components let's say 2 vectors and 3 D space I just take a Expedia plus citywide BY plus Azb easy and if that summer this comes to 0 the matter with the individual terms are is that some comes to 0 that means that the vector and the vector B are at right angles to each other and that's much much easier to more generally if we've got I'm lots of dimensions then we need to expand our son so it could be a 1 B 1 because we don't have x y and z if we've got let's say 5 or 6 20 we run our letter so we switched numbers were we won't run out they anyone B 1 plus a to B 2 was a 3 B 3 and which is right that the shorthand as the sun Over and over a and B. and that goes to however so are we want to go including in some cases to infinity and that should be 0 In the same idea holders functions except 1 the some becomes an integral because when you multiply the functions by so the they both depend on access so adding up you can just an you have to integrate To get the answer and number 2 because the functions can be complex we have to take the complex conjugate of the 1st function let's suppose we've got to functions F and G then are orthogonally conditions as follows the integral of F times G D X 0 now we can show based on theirs and the definition of her mediation but the Iden functions with different Oregon values are orthogonal so here's what we if it's an item function it has a Nikon value that's a real numbers so let's put all major on 5 1 and we get on May 2 1 5 1 we get so I went back because it's an I function we put Omega on-site do we don't make it too the only thing we need to know is that Omega 1 is not equal to all make it too they're different numbers they're real and the unequal and the operator big Omega had his her mission let's take the 1st item value equation and make a series of operations to both sides of that's always what you do when you simplifying equations you do the same thing to both sides methodically and if you do that you never get mixed up and nothing ever goes wrong and if you do some shorthand of cross blindness and then you don't know what you're doing exactly you'll oftentimes get it wrong the less take this equation Omega Psi 1 illegals on May 2 1 5 1 and was 1st multiply on the left we have to make sure we multiply on the same side when we do this by 5 2 star OK so now we've got 5 to star Omega 5 1 is equal to fight to stop little made a 1 5 1 and then since many was a constant I can pull it out and say that Little made 1 flight to now I'm gonna put an integral on both sides because of 2 things are equal that of and multiply them both by 5 2 star there still equal and on-site integrate them both over DX they're still equal they don't become unequal and so I integrate overflight too start acts major 5 1 DX and that's the goal of a major 1 instance on May 2 1 is a constant I pull it out and I end up with all make one-time interval fight to start by 1 the attacks and we can do the same series of operations exactly but instead of having no major Hatf III 1 we take Oh may have 5 2 we don't make it too and we just go through the same only we just what the roles of 1 and 2 we multiplied by 5 1 star and if we do that but I've just not done every step here but all major had fight to is equal to or make it too the considerable 5 1 star fight too now let's take the Konjic complex conjugate of both sides of the 1st equation so on the left-hand side I have the complex conjugate of the whole thing or made 2 all may have made 1 integrated and on the on the other side I have little major 1 at times in fight to stop fly 1 DAX
stock and I can simplified that I leave the other side along because that's going to be the definition from each of the right hand side I turned to a made 1 start and then the integral a 5 to star times 5 1 start while the start of the starlet's yeah change undermines our back on so that goes away and I can then write the 5 1 in front of the fight to it as matter of multiplying those there's no operator so I finally come to the following the goal of fight to start Omega had 5 1 is equal to or may go 1 times the interval of fly one-stop flights what does that get us while the observable as her Misha and so but when I put in Omega had fired to to give the item value or make it too and I do the same thing I find but I get the same thing backwards so all made it to Storrow made a 1 5 1 is equal to or make a one-star sorry 5 1 start Omega hat 5 2 and so using our 2 series of equations here's what we come to final all make it too times in Abilify violence .period fighting it is equal to a mail 1 time in goal of violence are fine but only to use not equal to major 1 so let's a track on May 2 1 times in a row from both sides then we find out that make it to top minus some 1 times the integral is equal to 0 but since make equal to or it must be that the other things 0 because if I have any number of times something the only way I can make the whole thing 0 is if the other thing 0 and that means that the integral 5 1 start 5 to DX is 0 and that means but they are orthogonal so that's that's the proof you'll have to go over a couple of times to get it down but that's kind of a standard thing that's done quantum mechanics to show but I can't functions for different values of Ivan values answer orthogonal now suppose we make a measurement of quantum system it's represented by a wave function that's not a nite instead state of the operator question what happened well we expressed side as a linear combination of right functions and that we know we can do that because there is no function that can escape us and we know our functions can be made normalized so we assume there normalized and then the probability of obtaining a particular item value let's say I value 10 a Out of the the total totality from 1 end is the absolute value of squared were K is the coefficients Of the case Eigen functions now suppose we then make the measurement again right away the question is do we get a different result and the answer is ,comma surprised but the answer is it turns out if we make the measurement again we get the same result and if we keep measuring the same observable over and over we keep getting the same results and now sort of mysteriously away it's 100 per cent certain that we're going to get that resolved there is no other result and we're going to get and I've tried to encapsulate this in this kind of pseudo equation we start now With probabilities could be any of these I can't stay and there we make a measurement In somehow 1 of them is chosen and we can't say how even in an ideal experiment but we can say with the probability of 25 per cent of the time we get this resolved now the measure again and nothing's intervene I haven't done anything we get the same result again and again and again and again and now there's no probability at all it's always 100 per cent of its exact certainty that is measurement is counter like a but all the other possibilities were filtered out leaving the 1 that's actually observed if you saw coins with the coin sorters you roll down and when they fit the size of of the of of the 2 they drop in and yet here you don't always too was going to drop into its 1st then that's like being uncertain but then if you drop the thing and it drops into the 3rd to the many empty it out there put it back in is going to drop in the 3rd to begin this kind of an analogy for what's going on here or we could say for example I suppose we flip a coin until it hits and start spinning and lies down I we assume its 50 percent probability heads and 50 per cent probability tale but once it lands and we look at it then we see it's heads I know that if we don't do anything we don't again just sit there its head its head again its head again and so forth and has as many times as we want keep looking at it and that's what this this theory of measurement as saying in other words when you make a measurement you rule out certain possibilities there now gone now the measurements made it came up that have become of this if you
make it again comes up this view make it again it comes up there and that's assuming you don't have any interaction in between but this is an idealized experiment we are talking about how we practically would implement and likewise in quantum mechanics it's just like looking at that calling if we make the same ideal
measurement again and again after filtering out all the other possibilities we just get the 1 result but we got the same result each time but before we made the measurement and it seemed like there were other possibilities and if we start all over not with the 1 we've measures but with an identical particle coming through that we haven't ventures there we might get a different answer and that if we measure that again will get that different answer again and so on and so forth and so it seems as if the measurement itself took this very fragile thing this way function and it made it collapsed onto Of particular Eigen function said right this is it and then all the other possibilities vanish forever if I decide I'm going to give a lecture I turn up and do the lecture but if I decide I'm going to the beach instead and I go to the beach than the lecture is not a possibility and has now vanished forever it's gone and I'm at the beach and so by making that choice lies I've narrow down the possibilities before I did that I can say about 50 50 I give a lecture or go to the beach and that's important because it means that measurement effects Kwan systems and that means that there is no such thing as a property without measuring it but we usually think that things have properties independent of measuring them because they seem to at this point for example has amassed whether I have it on a scale or not and I assume it's the same and for big objects that are always being bombarded by all kinds of things and never have a chance to let the wave functions to that's certainly true but for small things we have to be very wary about assuming that something has a property if we have not measured and because the measurement will change and so it could be that it was in some superposition Of the mixture and 1 measure we
picked out 1 of them but that doesn't mean it was like that before it means we might have changed so if we had obtained lets it go back to the coin if we have obtained tales instead on the 1st run and if we keep looking at this and so we 50 per cent probability and it collapses onto a particular choice have the the time and once it has collapsed on that particular choice it remains there for any number of repeated measurements at that does not now the question is this what happened to the uncertainty principle because now I'm claiming that we can get measured results with certainty we're saying we always get the same result which is measure once then the uncertainty goes away and that's kind of interesting because it's not so simple because the uncertainty principle which we quoted for position and momentum applies to measure and to think position and momentum at once or 1 right after the other not just 1 observable but there is no uncertainty about measuring 1 thing as well as you like the problem is if you want to measure what you think of everything but you could measure then there will be some problems some blurring perhaps that you didn't anticipate so a deeper analysis shows us that not all properties needed need to be answered In fact if the 2 operators at the same set of Eigen function this is why it's very important mathematically for us To be able to determine the Eigen functions of an operator because we might have this operator and that operator representing this and that and if it turns out that the 2 operators mathematically have the same set of functions even if they have different ideas about usually they will because they have different views and so on then we can measure both of them and we get exact results for both so we may measure this and that and we get a certain values and every measure this matter don't get the same for both and there's no answer but unfortunately position and momentum which 2 things that people like to determine we are not compatible and that way so there's this idea call complementarity observable incompatible cannot be measured to arbitrary positions will hear what I've shown it is a real coin I slipped didn't have become heads it's a penny and you can see that it's heads because you can see Lincoln in profile on the face of the coins and community other things on like the year meant but let's let's just say we can't help but its head now I suppose instead of trying to measure heads and by leaving their extended obviously never had had not gonna flip because I'm allowed to do anything to it except look at measure if on the other hand were interested in the exact segment as the Corning In that case we have oriented like this and this was tricky to do but the coin did balance on its edge it was thick enough from the surface was flat enough and the mine collaborator with steady offhand and now if you have the coin oriented like this you can see exactly how thick it is whereas when I was down with head pointing you have no idea I think it was imagining looking straight down on it they can get the best possible now you can try and now if if the coins on edge the thoughtlessly unstable and so any anything I try to do that is to look at it could have dropped the question is when it's like that which side his head and the answers because I'm looking at it John I have no idea which side it said and can't systems are very much like that if we have complementary variables if I try to 0 in on 1 of them it means the other 1 fades out and I can't get both at once because there interfering with each other and I was just there is no possibility of doing that we could try to cheat here's the coin bounced on pan with an eraser to keep it steady and we could look at the coins on an angle like this and the way it's angles here I can pretty much tell it's heads it's not as clear as it was before but I can pretty much tell it said but what I can do now is measured the thickness very well because I'm seeing the pictures from an angle and it gets smaller and smaller and smaller as a step the heads and then I can measure it very well and basically in order to get the thickness better I have turned the point toward me like that man finally at some point I can't see whether urgent bills now we're calling a macroscopic thing I can look at it had 2nd oriented and say Well that's IT was small things you can forget that that's not possible unless you can see it said you don't know what it is and that's the problem so the uncertainty principle really makes this but numerically rigorous it says exactly hello well you could measure the thickness of war and the ward tell its once it's a small things and when you know how the different variables the things you're trying to measure interact with each other that's basically what it's making it much more rigorous let's talk now about classical Adams those in a classical Adam we have Maxwell's equations and this was another problem actually at the turn of the century is that it was fairly easy to work out the head of an accelerating would radiate energy in accordance with Maxwell's equations but yes the electron which is certainly accelerating if we imagine it going in a circle around positively charged nucleus then it has to radiate energy than it has to slow down because the energy doesn't come from Nowhere and what that means is that the electron would spiral Kim toward the proton and would eventually condense onto it and if that happened there wouldn't be any electrons around to make bonds and so there wouldn't be any molecules there would be any life there wouldn't be any Adams even it would just be like a neutron star or something with just all this condensed matter somehow the electron it's not like interplanetary but and it's not behaving according to the way a charge would and Maxwell's equations and the reason it doesn't partly is the uncertainty principle because suppose the electron started slowing down and spiraling and coming in and then in smaller and smaller smaller what we just did a calculation that showed the electrons within 200 the commuters but the minimum uncertainty in velocity is like 100 thousand meters per 2nd and what that means is that it is impossible for the electron to spiraling and be on top of the proton in that itsy-bitsy space and the stationary because that violates the uncertainty principle and quantum mechanics it's not possible to measure position momentum that accurately and therefore the electrons may start spiraling and then maybe there's still gets tossed out
suddenly and go in a different direction and so on that saves so if the electron is not spiraling around like a planetary model of an In an orbit then what is it it's certainly maintains a stable probability distribution because we looked out and so we see that they have a cloud of negative charge around and doesn't change if we don't disturb the animal if we just leave it alone but we don't know where the electron is because the electron behaves like a wave unless we try to measure its position by which we would have to use a very energetic photons and blow the electron clean out of the basically and intercourse we've lost our threat we were trying to figure out what it looked like we didn't look at it the problem is that some you can only talk about the things you can measure you can talk about things that you can imagine what they might be stable distributions of the electron density have to be standing standing wave you can think of a guitar string if I put my finger set here to Camp moved here can move at the other end in between they can vibrate and it just makes a stable pattern and sits there doing the same thing and the electron then has to do something very much like them when it's in and it has a very interesting ways property we couldn't understand it at all if we thought of it as a rocker be fooling around on board that of course is a periodic trajectory but electrons don't have trajectory and so instead of on or bit we speak of orbital which is the wave analogy of a stable orbit now it's the wave function that has to somehow like the guitar string can only play certain notified it that France the wave function can only play certain notes in the adamant has to come round match itself and give a stable standing wave and that means the wavelength of the wave function has to match into the space into which the electron is confined if it doesn't you won't find the electron in that way function so there's destructive interference and the way function vanishes and if the wave function vanishes In the chance of seeing the electronic that energy also ventures because the wave function tells us the probability lets at 380 thing is kind of hard to visualize but we can certainly do it electron-ion to the sphere arrange and that makes it much easier for us to drop so let's have a look than on electron range here is an electron going around as a way I say going around but I don't know where it is because I have measured position but I have a standing wave it's equal everywhere in space and just on the real part when the real parts 0 you master part and that's why another reason why we have to have complex sometimes this 1 goes round and round and round and every time it comes around it's back in the same place goes around the back to the same place from Russia and so therefore it's going to make a stable repeating pattern that's going to sit there and if that we can't see anything going round and round eyes I imagined it was doing that as if it were a little rock going around and around the fact is just this past stable repeating that's because it exactly matches the condition that Mets need itself when it hooks up again but if I have a slightly different wavelengths so that it doesn't match when it comes around but it's a little bit off the goes around instead of matching perfect attendance higher then if it goes
around again it's a little worse the goes around again as little worse let's go up and down finally come around on the opposite side and let's go around and another time on this thing the 2nd time around it's the worst and if I go around several times it appears when I'd rather think that the waiver is up and down up and down up and down up and down everywhere and that means that cancels itself out there is no way it's up and down up and down the CIS canceled itself out the only position that can have a wave is the 1 that perfectly matches and there could be a higher 1 with instead of 3 loads of As for there are 4 that perfectly matches to but there's not three-and-a-half and there's not 3 . 1 and there's not pie loaves there is exactly an integer number of loaves and that means there are a certain number of integer energy levels that the electron can be and it can't just be any quick that's not allowed and that's very important because it explains the spectroscopic observations of Adams where they didn't just irradiate annual life but every element gave certain characteristic lines which depend where it was in the periodic table and so forth and so on and of course is very important for chemical analysis as 1 way you can tell what's in an unknown sample is atomic emission confined systems answer confined systems because the electron has to stay there but there are many other confined systems nanoparticles Sokol particle in a box which is a model problem we're going to do which is very easy mathematics compared real problem which is why we do it where every other confined system there's no matter how it's confined the way function has to somehow fit it has to fit into the space available if it goes round the bounces back and forth a does anything and it comes back difference that means that that particular way function is going to cancel itself out and it's just gone this matching conditions really restricts the wave function to a certain set of values and what gives us allowed energy states such as those that are close to that such that there are observed in atoms and molecules and form the basis of all kinds of chemical analysis that we're going to do if we've got much shorter wavelength life we learned that short wavelength equals high energy and the long-wavelength like equals low energy and to broadly said well but particles have a wave associated with them and now we've given the singer named the wave function and its assumption we can plot it if we have a functional form for which we sometimes do and we can look at and what we find is that if the wave function has higher curvature it's going up and down and down a lot like crazy that like a photon with a short wavelength and that means that that state that quantum state is higher energy than 1 that's all spread out and just kind of moping around not really very many up and down parts of up-and-down notes in in the thing but I'm gonna close here with the position operator and will pick this up next time we found that the momentum operator was too take minor part times derivatives the position operators in fact I gave it before is an example of an operator X have on is just equal to the number X on side and the number axis is is the position of of the particle the momentum might functions it is the VIP acts upon each part and to make sure that the I function is normalized we should include a normalization factor which I'll just put on some number here to make sure it's normalized and the question is what is the probability density look like for a momentum might instead well we just take 5 p time five-star off-peak times 5 feet we get an ETA the minus side PX and needed the plus side here even plus the the miners that's 1 because as he 0 doesn't matter whether it's a measure real still works we just get on square but that's weird because it says the probability density doesn't depend on it's just some number and what that means that is for a moment my said the particle as equal probability of being anywhere at all basically from plus or minus infinity equally likely and so position island states where the particle is definitely within 10 to the minus no matter how small you 1 thing at that point the momentum where the momentum is exactly determined 2 completely different aspects of measurement and there are completely at odds the best you can do it you can get the position to answer Ltd and mn simultaneously at the momentum within a certain limit but if you try to get too aggressive with 1 Sykes who use this then the other 1 gets wide because somehow this area between the 2 of them is like pushing on gel or something it squeezes out to try to get too aggressive with it and there's no way you can minimize that affect accepted to I have the very best uncertainty if there on the inequality but you can't make it 0 way liked devil position I can state on the other hand instead of this corkscrew should think of the acts as a corkscrew this in this way the particles going that way this quirk in the other way it's going this way but in either case it's just a constant thing cost around like a corkscrew driving through a wine bottle cork it's going to go a certain direction on the other hand position should be like that at all but it should be like this it should be but the wave function should be piled up like a big pile of sand in this position and then it should be 0 elsewhere because we know when we take the function square that tells us the probability of finding the particle to retake some function and piled up like the Eiffel Tower will see
particles can be there and then I might have some uncertainty might be out here but we can imagine piling up very steep and very high and that would be a position I can't function next time when I'm going to do is I'm going to take a model position function that's localized and I'm going to expanded in terms of momentum functions and show you that the momentum of such a function like that becomes more and more uncertain as we make the position sharper and then finally will finally introduced after the 1st week of classes over we're going to introduce the wave equation that's what we didn't have so far that tells us exactly how these wave functions move forward in time and how they have certain energy and other properties and that allows us then to discover what these functions actually are still pick it up there next time


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