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# Lecture 05. Model 1D Quantum Systems: The "Particle in a Box"

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the plus IPL upon each bar and even the minus I feel punished while the 1st equation just means that there is equal the minus B or B is equal the miner said the same thing and that means that we can write the 2nd equation as using a for both sides as a E to the IPO upon each bar minus to the miners IPL upon each bought and so we have the 2 acts as exponential functions have to be equal we don't want a to be 0 or then everything is 0 and the way function damages and therefore there is a condition that these 2 counter-rotating things be adding up to 0 at the edge of the box when taxes equal to L and what that condition is going to mean is that when these 2 corkscrews have developed 2 0 they can just be pointing out that they have to be pointing in opposite each other is going to mean that there is a restriction on the wavelength of the wave function and that means that only certain energies we are going to be allowed a very useful relationship here is Oilers formula which I've written out for you I think is equal to coast data plus I signed the if you haven't had a course in complex analysis you may not have encountered this kind of formula but it is in fact by far 1 of the most useful relations you can ever imagine and dig out just take a little aside here and show you how you can use this formula to drive all kinds of other formulas which some people try to commit to memory or they even write them on bits of paper and stick them on their desks so they will forget them as they're doing a lot of trigonometric problems the less do a practice problems let's use the above relationship EDI phase is equal to coast data plus I signed up to derive the double angle formula for the co-signed an assigned functions Of we have coast to the data we want to express that in terms of co-signed data and signed data if we have signed to say that we want express their terms of co-signed the inside and rather than memorizing some kind of formula that doesn't make much sense we're just going to use the properties of the exponential switcher very easy to remember and do a little bit of algebra which is quick about all we have to remember is that I square is equal to minus 1 and then we're in the clear but here's the answer or rather than memorizing the things here's what we're gonna do were going to set relationships and say the 2 the 2 I think is equal to the to the I favor square because that's how exponential work when you have something in the exponent and EDI status squared is to the I times I think on the left-hand side then we have co-signed to theta plus I signed to think because that's the 2 the 2 I think or III to VI to think if you like on the right hand side where we have the I think that at times you the I'd say we have co-signed plus I signed times "quotation mark plus I sign and if we work that out on the right hand side that's co-signed squared data plus 2 I co-signed the signed data plus I squared signed and at this point all we have to remember is that I squared is sequel to minus 1 and therefore we end up with cosigns squared data plus 2 I signed politicos data -minus sign squared fade so what we have co-signed 2 plus I signed to fail on 1 side and we have this thing with 3 terms on the other side well these are too complex number and we're saying these 2 complex numbers are equal and therefore the key thing is that that means the real parts a equal and the imaginary parts are equal because the way we think of complex numbers as the real parts along the axis and the imaginary parts along wide and as the 2 numbers are equal they have to have the same value of X and the same value of wine while the real power on the left-hand side is co-signed to fade the real power on the right-hand side is coast square data minus signs "quotation mark data that's the co-signed double angle formula the imaginary part on the left-hand side is assigned to the the imaginary part on the right-hand side is to CO status signed there and therefore that is the relationship between this sign of 2 and the signing co-signer status and it's very quick and easy to do stuff like this you don't have to remember anything you go work out the triple angle formula you go about this and that the other very easy and quickly it's much much much better than
memorizing formulas that you don't quite fully understand if you rely on your memory you might get it wrong and for assured you get older you'll start making more and more mistakes if you just read arrive at every time it's very easy to do and much more reliable so let me just close by saying any trigonometric identity that you've seen can be derived by using these complex exponential stand just relating the real imaginary part without any any particular other talent OK let's get back to our particle on a box now we know that the wave function vanishes at the left at the right hand side of the box big and we put in those numbers we had a 82 the plus side pH I P L upon age bar minus 82 -minus on and therefore we just put in "quotation mark close I sign and we get a Times co-signed PL upon each bar plus I sign minus co-signed plus I signed and therefore at the end we get to I L excuse me too I a sign Pl upon H that's the condition but that that thing equal to 0 they can be 0 I is definitely not 0 2 squared of minus 1 2 is not 0 and therefore the sign function has to be 0 the sign function is 0 0 when the argument the data which are set to be equal PRI over age bar it is an integer multiple of of Pa and we work in radians we don't work and degrees we always put things in in radiance dimensional and the integer values that are allowed for sale are equals 1 2 3 and so forth and therefore Pl Over over-age Park should be times pie where and has some positive integer again we can't have any equals 0 if we have an equal 0 then the sign function vanishes everywhere and that means the wave function that ship I therefore what we found here in this little bit of mathematics is that the condition of the particle be localized that it be restricted to some region in space and not just allowed to go wherever it wants results right away In the result the Met momentum and the energy Of the particle we are quantized they can only occur In set amounts we can take our resolve a step further now by doing the following let's solve for this quantized here's what we find the momentum is equal to empire H-bomb Over and since age power is aged over to pie then the momentum is just an age over to and that's just using the definition of H bar that means the energy is quantized because the particle inside the box only has Connecticut Energy Peace Corps over 2 AM and if the momentum quantifies the museum energies quantized and the energy Of the nth allowed state that can be occupied is then squared age squared over 80 ML square just by squaring the momentum and dividing it by 2 L this is an important formula for a couple of reasons 1st it shows you how the quantized edition is occurring 2nd it shows you that the latter of states is going up like the squares not evenly not getting closer and closer together and what will find in the future and the future lectures that that has to do a lot with the shape of the potential so if we imagine that the potential is written as a power series we could write via X is equal to something time sex square and then we can say it's about X to the fore we could make it higher and higher and if we made it next to the Infiniti it would be 0 1 X is less 1 to sail inside the box then would suddenly blow up to infinity the 2nd you go higher than that and and that fact that the power that of the potential is very high is the reason why the states get farther and farther apart if the power's slower than maybe evenly spaced or they may get closer and closer together putting everything together now In what we've got we've figured out that everything's quantise we know the former the wave function and therefore we can write scion is too I capital sign PX upon each bar and then we can express that In terms of our quantise Asian conditions using the quantized momentum and high Annex over L so now knowledge bars out of it and we just have this nice wave in the bar as the quantum number increases the number of nodes in the way is going to
increase and that is the condition that the energy increase amounts of course exactly what we expected but it's nice that it just comes out and that it makes perfect sense but in fact what we're gonna find in the future is that the so-called knows the places where the wave function crosses 0 are ever so important because they give us a really strong clue as to what kind of behavior were going to see where you have a node in the wave function that means that the particle is never found there because sigh is 0 so star it is 0 and the particles never found there ineffective it crosses through 0 when you squared it gets very small because it's smaller negative on 1 side and small and positive on the other side suggested the probability smoothly goes to 0 and that means although the boxes uniform there are places in the box where we do not ever expect to find the particle if we try to measure its position and that's certainly seems different than any classical situations that we might think about where we have a ball inside a box and the ball could be anywhere in error of course the quantity state of such a system has huge and and we could never see this kind of various social behavior because the broadly wavelength of the ball is far too short well we still need to determine capital he went away because it was related air and the only way we can determine a is by normalizing the wave function recalled that normalization just means that we insist if this way function is going to be measuring probability that the probability that the particle be somewhere in this case somewhere inside the box is 100 per cent for unity we can't have it be less than that we can have been more than the let's go ahead and normalize 3rd the particle in a box Energy Icahn functions was will will do the ground state but it's the same for the others and this will be practiced problem not but this is a simple calculus problems however before you do any kind of integration what you should check is whether you
can do anything to to the grand to make the integration easier and we certainly can't because the easiest function in a great or differentiate is the exponential function and if we can our the problem in terms of that then were out of the woods if we use some other functions where we have to play around a lot we don't know the Yantai derivatives than that could be problematic no here's what we need to have the integral from minus infinity To infinity of size star sign this 1 but the wave function is 0 outside the box and so in this case what we can do is simplified the integral To the integral from 0 to because that's the only 0 0 otherwise so I stop sign is equal 2 1 and in terms of our on normalized what we have is the integral member if we have and I we have to put the minus side so we have minus 2 IEA sign and acts upon times to IA and Pyrex upon L minus ii time size plus 1 of course that's why we always take the complex conjugate to make sure that we get a real positive probability a doesn't depend on access and therefore we can pull it out of the unit and we get for 80 squared times the integral from 0 to L. of signs squared In pirates upon and we want that to to be 1 and once we find out what the value of the integral is we can work out what areas and then we can go ahead and put a into our formula for the particle a box wave function and there were done with the whole 9 yards now you could look up the integral In the book that's what I used to do when I was a student often if I couldn't do the integral you could use software like Mathematica math Qatar's something else maple or you could look it up on illegal stock ,comma but just for an exercise here I want to do the problem just do it right through and I wanna do it by changing they side too Back to EDI because when derivative or an integral I think maybe but turned it back to the exponential this case is going to be much easier for me to do it and the only Formula I need here is that the role of the DAX DX is equal to 1 over a currency explores some constant before doing a definite integral the constant is going to go away said subtracts out when we take the 2 limits of course and that's all I need to know and the only thing I need to know about complex numbers is complex numbers behave the same way as real numbers but if we have eaten some funny complex thing it's this funny complex thing and the denominator and funny complex thing again and you just close your eyes and keep going let's have a look we can write a normalization conditions like this rather than the 2 I signed we used to eat the even the plus side you might as well and then the same thing even plus ii the minus side times the integral and that's equal to 1 and that simplifies to this then we end up with a squared times integral from 0 to l of 1 minus the 2 them -minus heeded the I might CDI plus 1 and that should be equal 2 1 now the complex exponential census Siegel echoes that of course I signed data it you square and you have the square in the formula there then each of the 2 I thought I is equal to 1 and therefore that part goes away when we evaluated 0 now both conditions it goes away because of the way the wavelength of the wave function is set and so we end up with 3 terms here which I've written out but should be equal to 180 squared and then to access the enter derivative of 2 evaluated at L and 0 that part has has to work out and then the other 2 parts that have won over minus 2 IA because that was in the complex exponential here where I've let ages be the collection of Constance P & H Bomb and so forth I and then another term and the ones with the 2 and the denominator fortunately for us we don't have to worry about because when we evaluated it's 1 as well and it's 1 and 0 and so when we subtract those limits that part just vanishes From the equation and were only last where the 1st term which is too well minus 0 for 2 dimes 0 and therefore they squared times that should be equal to 1 so finally at the end of the day we find out that capital there it is equal to 1 all over the 2 times to square the square root of 2 of but here's our final wave functions so I N of axes to IA and that brings the 2 from the bottom to the top and so the time sign pirates over all and that's equal to i Times Square to upon our sign and Pyrex over that would be our final result except we still have a bias we don't like the other it's still bugs people we've got an imaginary wave function it doesn't change the probability distribution it does not change the energy it does not change anything to multiply away function by a number that's complex that has unit length where
you should think about if you're multiplying away function by a complex number with the unit length is something like you're changing the color of the when you are changing the shape of the and since when you take the complex conjugate you get the opposite color when you crash and together you always get the same thing and for that reason at this point we can get rid of the eye and that's conventional mass cull picking the face of the wave functions were free to choose the phase to be whatever we like we don't have to have an eye we can have a 1 could have a minus ii doesn't matter but just to make it a little bit easier to understand if your 1st encountering the conventional choices to choose the states to just be 1 and when you pick that face you get the final solution here sigh and of X is equal to the square of 2 overall time sign empire Texas Nova fell the lowest energy state for particle is an equals 1 because remember we can't have any equals 0 or the wave function vanishes everywhere America's nothing anywhere there's no particle and there is a no-load at each side of the box but nowhere else so here's a plot of what the ground state wave function looks like it's just half a sine wave comes up reaches an apex in the center of the box which is apparently the most likely place to and then goes down toward the edge of the box and what we see is that even tho there's no potential at all in the box the fact that there is a potential setting limits I really profoundly changes with the wave function does it makes it far less likely that it's going to be near the edge of the box it's as if the wave function not has a 6th sense of knows and I don't want to get too close to the box because the energy out outside the box is infinite like putting your hand on a the wall and detecting that there's fire in the next apartment over and then you decide to stay away from the wall and that's pretty much what the wave functions doing here and remember what were plotting here is the probability amplitudes we have to take this thing that we've got and square in order to figure out the probability density and when we do that we find that the edges are really very low because they're going up like ax another EC squared that's even much smaller comes up and then sort repeat the middle and comes down again toward the other it's so let let me up no take this wave function we have an explicit for the expect women explicit form the wave function lets figure out that the expectation value for the position and the momentum Of the particle In this ground energy I state we know it's not in a position I had stayed because position I stay means there's 1 position and this has a range of positions it is in a moment of I state but it's a superposition of 2 possible values of p so it's not actually in the momentum state it's in a superposition 2 equal and opposite moment against let's then go ahead and take this and just work through as an exercise in expectation values and we're going to have to do some challenging and girls because we always have to do the integral to do the expectation value and in addition were going to have to do some derivatives so it seems that were going forward backward because remember the momentum operator is the derivative times I each and therefore we're going to have to do both so recall here's our formula for the expectation value of any operator the expectation value of some operator Omega is equal to the integral of side Omega had operating on side DX for our finite box and for the position operator we then have the following thank the expectation value position the average value after a large number of measurement is equal to the integral From 0 L I love squares to upon L signed by IX apart upon all times X because remember from the X hat operator just multiplies by X the number that the variable so the hat goes away we just have access and we and the other one's sense of both real so I and star are the same and we can do that integral easily and the expectation value comes out to L. over 2 which is perfect the most the average of all the possible measurements of the box averages to the middle what would have to because the boxes cemetery how could be anywhere else but at least it's interesting that the map just tells us what we already knew How about the momentum operator this is a little bit harder the words work through this the expectation value of P is equal to the interval of size .period he signed DX and that we have the derivative function in here and an integral and if we Clark through it we have to take the derivative of sign Taiex upon and that gives us a co-sign Amendment also brings out 1 over L so we and we end up with Indiana -minus 2 I bar comes in goal from 0 to L of sine die some extra stuff that goes on and this isn't so easy to figure out but it turns out that we could rearrange it keep in
mind however that the integral has Simon co-signed unit it doesn't have anything else it doesn't have any imaginary part and we know because the momentum operator is Hermitian operators that the expectation value has to be real and we have a big out in France and the only way I time something can be real is if the something it is 0 None of the other things hello and so forth have any imaginary part and therefore we could just use that argument to say that the expectation value of the momentum is 0 and how would make perfect sense because the particle is not on average going anywhere it's trapped in a box so how could it have anything except the expectations of the the moment of the Xerox the I'm now is a further illustration let's go forward and calculate the values of EC squared and Peace Square you'll see in a minute why were bothering To do that but the expectation value of x squared is the same thing we have the integral again and now we have the integral from 0 to L of EC squared signed squared Pyrex upon all DX with 2 over all of France then again you can either do this by parts you can do this by software you can look it up and what you find is that the expectation value of x squared the square of the position comes out to be L squared over 3 -minus elsewhere Over 2 times price that's what the integral works out you can probably see that the 1st part of the all-square over 3 comes from integrating X squared times 1 and the other part comes from integrating the trig functions a couple of times so that tells you pretty much how that's coming about but that's the expectation value of the square of the position that's not in the middle and it's certainly not 0 for the expectation value of Peace where we have to take to derivatives and then do the integral we have minor age borrow the derivative sides bar the derivatives would take the derivative once the sun goes to co-sign we take the derivative again the
co-signed goes to minus the sign conveniently canceling out the -minus NYTimes minus ii which is minus because I times size plus 1 so minus items times Midas Eyes minus 1 and all goes away and of course we expect the expectation value of something squared to be positive as cannot to be negative or something else we would be an indication that we made a mistake in the algebra and therefore the expectation value peace squared turns out to be 8 squared and Over elsewhere we can combine these 2 then then some interesting little exercise why would this be interesting In the answer is we had the uncertainty principle and the uncertainty principle applies to any wave functions if we measure the position momentum the uncertainty in them has to satisfy bigger than are equal to each borrowed over to and now we can figure out what that is and in doing so we can really expand on what this Delta X Delta p me let's think of the variants which is the deviation From the meeting squared Of the measurements the expectation value of the variance is the expectation value of backs minus the expectation value of X also square so the expectation value of that well if I ride out X minus the expectation value of x keep in mind Ex it is a variable here then 6 expectation value of X is just a number and when we work it out then if we work it out we get the expectation value of x squared minus 2 x times expectation value of X plus the expectation value of X square the expectation value of 2 x times expectation value of acts it is just too hence the expectation value of X because the expectation value of the expectation value is just what you expect it's just the expectation right of access and therefore the 2nd term subtracts from the last term and we get but the variance is equal to the expectation value of x squared minus the expectation value of X square and the same thing applies for the momentum the variance in the momentum is the expectation value of the momentum squared minus the expectation value of the moment square if the distribution is very narrow yeah but in fact they all have the same momentum then the expectation value of the square is equal to the square of the expectation value and the variances 0 and same thing of 4 position if there's no spread position than the variances 0 that just means that it's a very very very narrow distribution could have an exam where every student gets the means I have no variance at all and would be very hard for them to make a curve for that class the variance in possession while we knew the expectation value was l over too so we're going to subtract that square and we worked out the expectation value that squared by doing that somewhat nasty integral those elsewhere over 3 miners elsewhere over to and that works out with a little bit of algebra To be squared times 1 of 12 miners 1 over to price where this doesn't want to encouraging because it usually things a little bit neater than that but I that's another good reason to do this problem because things are always so need as they might be in and some set up problem the variance in the momentum while the expectation value the momentum was 0 so that's easy so the expect the variance in the momentum is just the expectation value of Peace Square and we got that worked out a square post Worrell square as you can see and therefore if we associate the uncertainty in position and momentum With the square root of the variance the variance as the unit square length and square momentum if we take the square root of that we get a measure of the width of the and for the uncertainty and possession that then becomes a it has to do with the box the bigger the boxes the bigger Delta exits will that makes sense because the particle can spread out of the box ghostly nearly an hour as a square root 112 minus 1 over 2 square whatever that and for the momentum we take the square root and we get the formula shown now we had a relationship the uncertainty principle it said Delta X Delta P is greater than or equal to H bar over to let's have a look if we take Delta H Delta p we did the following formula and the key here is that when we simplify it and pull out the pilot to get rid of the power that we get the following thing we have high squared minus 6 In their at times 1 3rd in the square root so we got our age bar to and then we've got square root of one-third times pie squared minus and piste where it is I don't know what it is but it's bigger than 3 squared because I do know the pies bigger than 3 so that thing there whatever it is this is bigger than over to times once times 3 squared minus 6 rather than password minus 6 and that it is Justice Barbara too and so what we've shown is that for a particle on a box the uncertainty is bigger the minimum uncertainty the Heisenberg did never say that the uncertainty imposition times momentum can be should be equal to the minimum In fact that's rare usually you can even do as well as that but what he did say is that you can never be last the minimum and thank goodness will work it out doesn't depend on the length of the box the uncertainty principle because of the box gets longer than the momentum becomes narrower as the position gets wider and therefore what we've shown here in town simple calculation which did take a little bit of time but nevertheless it's not is fairly straightforward is that for the ground state particle in box that we satisfied the uncertainty principle in fact were about and at least 10 per cent to compared to the absolute minimum that we could have and if we did the same calculation for the and equals 2 and it costs Fredy whatever state it would be the same conclusion exactly is that we would satisfy the uncertainty principle for all those states and if you want to challenge go ahead and try for something other than the ground state to calculate Delta the Delta acts the same way we did but to substitute a new new formula and convince yourself that it also always satisfies inserted the principal the energy of the particle increases like and square and the wave function has more notes for higher energy and what you what you often see in books is sort of a complicit plot and these complicit plots can be confusing when you 1st encounter them because in actual fact what we're doing is planning 2 things at once we usually bought a flat line like a ladder to indicate this energy the particle can be at this and we have these things going up like an square and then To give you an idea of what the
wave function of the particle looks like we thought the way function of the particle on the as if the line were 0 even were moving the lineup in energy and do you have to get used to these plots because if you if you think of them in terms of energy around planning something weighty here that's going up and down in energy you're missing the whole point of the energy is given by the line and then separately we're just plotting the wave function to see what it looks like it doesn't have units of energy and we just plodded on the same but Graf because we're lazy and somebody did that early on and that everybody liked it once they get used to it but when you 1st encounter these kind of plots they can be very confusing you just don't finally don't know what's being plotted here is a typical plot them but you might see in a book I've just taken the energy scaled as and square the 1st energy level is 1 square it's 1 2 square is for 3 squares 9 and then on those energy levels treating them basically is the 0 I've plotted the 1st when function which I've called so I 1 that's the ground state and then the next 1 which has 1 node in the middle but that 1 of Lloyd's being in the middle the bottom 1 likes being in the middle of the next 1 up avoids being in the middle and then the 3rd 1 I 3 which has 2 nodes in the interior of the box and that 1 likes being in the middle again and likes being closer to the edge of the box and if you take very very very high levels of banned then you'll find that you just get these little ripples everywhere and basically the probability of being anywhere is basically on the same there there's little things like an egg carton but those things are so tiny tinier than the width of the nuclear on and that there is no way we can do an experiment to try to uncover that kind of core edition in the way functions OK I want to close their and ask the following question which will get to in the next lecture and that has to do with a phenomenon called tunneling we made the potential go to infinity at the edge of the box and the reason we did that is you're going to see what happens if we don't do that which is that we get set up a lot of math problems to do and not all of them are easy but what would happen if instead of being infinite it would be edges just Rosa To some finite value instead what would happen if we had a plot where we have the potential be higher then the energy of the particles so classically the particle is trapped and has to remain within the wall forever but it doesn't go to infinity and therefore we can't use the argument that size should necessarily vanished completely because he is a small part of sigh leaks through and that adds a high energy from the potential being so high that doesn't necessarily mean that all bets are off because by spreading out more so I has had not so much curvature so it could be that it can relaxes pushes out slips out a little bit and that's a better situation than having it cut off just because of the edge of the box and in fact what we're gonna find next time is that because of this way nature that the the way function always sneaks in the sum forbidden region it's sort of like somebody at 4 AM rolling through a stop sign when nobody else is around and there's so it's very interesting in fact that there's a big difference between classical mechanics and quantum mechanics because in quantum mechanics there always some chance of escaping out of jail no matter how type the bars are and eventually in things like radioactive decay for example that's exactly what happens so pick up quantum tunneling in the next and the next
Biologisches Material
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Chemische Forschung
Tiermodell
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Krankheit
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Falle <Kohlenwasserstofflagerstätte>
Derivatisierung
Aktionspotenzial
Nanopartikel
Nanopartikel
Krankheit
Wasserwelle
Funktionelle Gruppe
Lösung
Proteinglutamin-Glutamyltransferase <Proteinglutamin-gamma-glutamyltransferase>
Komplexbildungsreaktion
Mineralbildung
Fülle <Speise>
Phasengleichgewicht
Restriktionsenzym
Memory-Effekt
Potenz <Homöopathie>
Küstengebiet
Sense
Chemische Eigenschaft
Chemische Formel
Chemische Formel
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Infiltrationsanästhesie
Komplexbildungsreaktion
Memory-Effekt
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Gangart <Erzlagerstätte>
Konkrement <Innere Medizin>
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Metastase
Aktionspotenzial
Azokupplung
Konkrement <Innere Medizin>
Altern
Sense
Bukett <Wein>
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Nanopartikel
Chemische Formel
Alkoholgehalt
Krankheit
Wasserwelle
Funktionelle Gruppe
Single electron transfer
Phasengleichgewicht
Feuer
Fleischerin
Sonnenschutzmittel
Lösung
Aktionspotenzial
VSEPR-Modell
Altern
Stockfisch
Derivatisierung
Repetitive DNS
Sense
Nanopartikel
Sammler <Technik>
Chemische Formel
Operon
Funktionelle Gruppe
Wasserwelle
Komplexbildungsreaktion
Konjugate
Phasengleichgewicht
Fülle <Speise>
Potenz <Homöopathie>
Querprofil
Komplexbildungsreaktion
Operon
Konkrement <Innere Medizin>
Eisenherstellung
Chemische Formel
Thermoformen
Nanopartikel
Farbenindustrie
Krankheit
Zelldifferenzierung
Proteinglutamin-Glutamyltransferase <Proteinglutamin-gamma-glutamyltransferase>
Periodate
Azokupplung
Altern
Derivatisierung
Sense
Derivatisierung
Nanopartikel
Flussbett
Sonnenschutzmittel
Operon
Operon
Mineralbildung
Leckage
BET-Methode
Muskelrelaxans
Konkrement <Innere Medizin>
Aktionspotenzial
Teststreifen
Altern
Sense
Nanopartikel
Amrinon
Delta
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Terminations-Codon
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Setzen <Verfahrenstechnik>
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Valin