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Extra credit, very super hard! Consider set of N IaSSCS mwhich are still all equaL, but COHectexl by springs which alternate M Valuc: k.2k,k,2k.k Compute tic nOrIA]...

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Extra credit, very super hard! Consider set of N IaSSCS mwhich are still all equaL, but COHectexl by springs which alternate M Valuc: k.2k,k,2k.k Compute tic nOrIA] modes Ac hence the relation betWecn ald q. You will itucd (it'$ hard?) tat there are actually two typGs of modes. One is calledl acoustic phonons' aId the other is cllc| opticnl phonons' The algehra in this problem is also sitpl Model ol |d g"p)S for electrons HOving i solid like polvacetylene (a dimerizee chain o

Extra credit, very super hard! Consider set of N IaSSCS mwhich are still all equaL, but COHectexl by springs which alternate M Valuc: k.2k,k,2k.k Compute tic nOrIA] modes Ac hence the relation betWecn ald q. You will itucd (it'$ hard?) tat there are actually two typGs of modes. One is calledl acoustic phonons' aId the other is cllc| opticnl phonons' The algehra in this problem is also sitpl Model ol |d g"p)S for electrons HOving i solid like polvacetylene (a dimerizee chain of carhon atoms) wlcre the distances hetWecn aljacemt carhon atoms alternates 6om short t0 long; AIlc Ience (e ability ol electrons (0 'hop from carbon t0 carbon alternates betwevn casy and hard: So it's nice problem t0 master.



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In the following quesrions rwo statcmenrs (Asscrtion) (A) and Rcason (R) are givcn. Mark (a) If borh $A$ and $R$ arc corrcct and $R$ is the corcct cxplanation of $A$ (b) If both $A$ and $R$ arc corrccr bur $R$ is nor the corrcer cxplanation of $A$ (c) $\mathrm{A}$ is rruc bur $\mathrm{R}$ is false (d) $\mathrm{A}$ is false bur $\mathrm{R}$ is rruc. (c) $\mathrm{A}$ and $\mathrm{R}$ horh are false (A): Bond ordcr in a molccule can assume any valuc, positivc or ncgarive, inrcgral or fractional, including zcro. (R): Ir dcpcnds upon the numbcr of clectrons in the bonding and antibonding orbitals.

For this problem on the topic of quantum mechanics, were to consider a one dimensional lattice of carbon nuclei, which has entered into atomic spacing of 0.5 nm. The potential energy of one nucleus due to electrostatic interaction with its two nearest neighbors is given by the equation VfX is equal to accuse spread over four pipes were not into one over d minus X plus one over D plus X. Where the charge Q. is six e. And that's the charge of each carbon nucleus with exits position loaded to the equilibrium position. I want to first sketch the potential V X from X is equal to minus lead to X. Is equal to D. We have been told that at low energy nucleus remains near its equilibrium position and the potential is approximated as a harmonic oscillator. We want to then write VfX is a quadratic function in X and drop the higher order terms. We don't want to find the effect of spring constant K prime. The energy of the classical ground state and electron falls and the energy of the lowest energy photons emitted when the system transitions between energy levels. And lastly, the wavelength of this corresponding radiation. So here we have drawn the sketch of V X versus X as X varies between minus D and D. Which are about a sandals. Now, for part B. If we combine the terms in the given equation for vfx, we obtain That this potential function, the effects is equal to two K Q squid of the D times one over one minus X over D squared. Where K here is one of the four pi. Absolutely not. Now. If we expand the right hand fraction would one plus zed all to the power end is approximately one. Bless NZ. Where Z is much less than one and is equal to -1 and Z is equal to minus X over the old squared. We get the potential function vfx to be two K Q squid divided by D into one plus X squared over D squared. Now, if we use the charge Q two B six E and you put in K, we get vfx to be 18 E squared divided by pi absolutely not time's D plus 18 E squared divided by pie. Absolutely not D cubed times X squared Mhm. And so we have the potential function as a quadratic function. Now, for part C, we want the effect of spring constant and we'll use the second term for vfx and we get V is equal to a half K prime X squared where K prime is the effect of spring constant. And the second term is 18 E squared over pie. Absolutely not D cubed times X squared. So that means we can read off the effective spring constant. Care prime to be 36 E squared over pie. Absolutely not D cubed And putting in our values. This is 265 Newton's Panetta. And so this is the fossil the spring constant of the quantum harmonic oscillator novel party. The classical ground state is that X equals to zero. And so the energy at the stage at the state V is equal to 18 E squared over pie. Absolutely not times D. And this is if you put our values in 207 electron volts. Now for part E we want the energy of the lowest energy photon. Use the using the quantum energy states we have, the note is equal to a half H bar times the square root of K. Prime over em. And E N is equal to n plus a half H bar times the square root of K prime over em. So this implies that E. N is equal to two. He not into end plus a half. Now the lowest energy photon is due to a transition from the N. Is equal to one state to the N. Is equal to zero state. So therefore the energy transition dot E is equal to two. E note into one plus a half minus zero plus a half, which is two, the note or two into zero point zero 379 ElectronV. And so we get this energy transition of the lowest energy photon. It could be a zero 075 it electron volts. So he lowers energy photon has this energy. Lastly, for part F, we want to find the wavelength of this four time. We know I. E. He is able to hc over lambda and with the E being 0.758 electron volts, We can rearrange and solve the wavelength λ. We get λ to be 16 0.4 micrometers.

41.34. There. These, uh, two things were supposed to show that they're true. This has to do with We have, uh, undocked semiconductor. We could have a simplified model where we replaced the actual distribution of energy states, uh, with just having and the states in the violence ban that I'll have the same energy and see states in the conduction band that I'll have the same energy. And given that, uh, and if the number of conduction electrons is equal to the number of violence holes, we need to show that this is true. And the next part if we have going from from this if the for me energy is in the gap and the difference between the Fermi Energy and the two the conduction in violence energies is much larger than the thermal energy. We can derive this expression. So the number of electrons and the violence ban is this end the so that the density of Vaillant states time is the occupation probability. Uh, right. Okay. Theme I c f over Katie. Plus one. Now the surveillance in the violence band. We want the holes. So the number of holes in the violence band, Since the number of available states has to add up to the number of available states, we can subtract this expression we just found from envy and lend up with this. This is one of the things that is supposed to be equal to the other. This is Delta. So so far things are looking good of the number of electrons and the conduction, Ban has again equal the density of states times the occupation probability, exponential of difference for the conduction band thermal energy plus one. So since these two things are supposed to be equal to each other, these two things are equal. And that's what we were asked to show over heart beat. Here we're saying that the, uh, energy difference between the conduction and balance energies and the Fermi energy are large. And the Fermi energy is in the gap between the two and so the exponential zehr going to be much greater than one which lets us right slightly simplified form where we've left out the plus ones in the denominators because they're negligible compared to the gigantic value of exponential Sze Good. So recall that, uh, this is he c minus the Fermi Energy. This is the Fermi Energy minus V. So we can rearrange. This is equal to the ratio of our numbers of states. So we take the longer them of both sides, and that gives us our log of envy over and see. Um, we have this The difference here that becomes some of these two that also goes over here. And then we divide by two. And just like we're supposed to be, find that the Fermi Energy 1/2 claims the conduct INTs and violence energies plus this Katie log envy over and see. So now we want to show that the Fermi Energy is near the center of the gap when envy and NCR roughly equal, and we could see here that they're going to go. This is this ratio is going to be one the logger them of 10 And so we're just left with the average value of the two energies which will be directly in between her directly in the middle of the gap between Okay,

Okay. Hi friends in this problem the equation of motion. Okay. Yeah. For the be it must right in that chain edge X table dot be but that's too omega notice square XP minus may go unnoticed. Square X people is buddy minus X p minus but It's called 20 for Peace Corps to one to end four p. We have this special case acceptable dot n Plus two. Omega not squared accent minus omega notice square X And -1 is cool to me. Got not a square edge cause of mega tv XP is called to a P cause of omega T is called to see sign of and for people costs of omega T in equation but right, we can write minus a p. Omega is square and that's to Megan not square ap minus omega naughty square A people? S but minus a p minus pond. It's called 20. So we can write ap into omega not square minus omega squared squared too. Omega not squared A people S one, but as A B -1. So finally ever result can be simplified as to omega not square minus omega is square. And to see sign off at four P, it's called to omega notice square Sign of L for into people. S one plus sign off, I'll find to be minus. But so have a big question, we can right to omega not square minus omega square times see sign off, add far into pain. His call to omega not a square see sign off and find two P plus one plus sign of L fab b minus. But And this equation can be simplified for calculating alpha and alpha. You will get cause and yourself 1 -7. Guys squared upon to omega not the final mask amplitude is equal to for the final mosque, the amplitude is equal to don't He? End Plus one escort to see times a sign off. I'll find to end plus one. It's called to edge soc sculpture. You will get edge upon sign of L fine tube and place, but they told Thanks for watching it.

Bella dance are in one dimensional read books with infinitely high potential walls so we can calculate the energy off and in liberal by the relation e. And it is ical toe. So get up and into energy. Obviously will and energy obviously will is equal to so get up belongs constant derided by and into a mess up in the wrong times. It's good off lend off books. I became a typical cleared alevis level energy by putting different values. Here s Plank's constant is 65 or 63 die intended of power Minister before Jules again derided by anting toe mess up with Ernest nine point 11 Time attended a power minus 31 Get a good arm and lent out boxes five time tenet of our minus tan meter. This gives us the one equal to do you want for one time 10 to the power minus 19 You're which is a call to one point 51 electron. What now again, Is it a calculated the energy off higher liberals. So you do that is it going toe? So get off to Indo U on the physical affording to anyone he everyone is 1.51 Electron world This gives us e to equal toe 6.0 for electron What? Now we have the three equal toe So get off free and do you on Which is it called 09 into 1.51 is down world. This gives us the three equal to 13 point five nine Alec Don what similarly we have before equal do so get off for into everyone This gives our 16 times 1.51 electron world that is it going to 24.16 Electron world So we didn t We have the energy off 50 level that is you five equals Do if I have secure times He will this you sister and do five times 1.51 electron world equal to 37 point 75 electron world and easy's is it will do so Get off 16 to you one day Excuse us 36 in tow He won these 1.51 electron world So this gives us even e six equal to 54 Quite 36 electron Walt Now we can easily draw imaginable diagram by showing different electrons. India respectfully Waas. So this one is any cool? Do one having energy off 1.51 electron world. Next we have and equal to two having energy off six. Quiet Do four electron world next year an equal to three at energy Oh, 13.59 electron world. Next, we have an equal do for energy off 24.16 electron world. Then we have an equal do five at energy off for the seven point 75 Philip turn. Well, then we have unequal toe. Six at energy will equals toe 54 wind 36 Electron world Yes, it is given in the problem that there are two electrons in an equal to one one good been off and the other one with spin Don't is that I don't assign electron sodding excited. It's a dead and the third electron is in an equal do six. So by using the selection rule giving the problem that is delta and equal to number we have the first transition from an equal to five to any court to an equal to 62 physical through five. So from any cool too 62 and he called to fire in this case Delta II musical to one, which is what? So this transition is possible. The next transition is from an equal to 62. And is it good toe three from any gold to 62 and he's going to three in this case again. The time is he going toe. See which is odd. So this transition is also possible. Love again. Calculate. We're lent off immediate for town doing these transitions. So from an equal to five to and it will do Saudi from an equal do six door and equal do five we have. We're lent up immediate for down equal to Plank's constant times. CPT oblige. Derided by energy, is Plank's constant. I am sip it Applied is equal to 1240 electron world, no matter derided by and not your foot down, which is actually called to and are the difference between and equal to six and an equal to five. So energy off any gold? Two sixes, 54 wide. 36 Other turn world minus energy off and five is that is seven point 75 Don world. This gives us I lent up in beautiful down during this transition equal. Do 75. Animated Now for the next transition, that is, from any call to 62 an equal to three. We have learned equal toe plants. Costin Indisputable I divided by energy. Planck's constant inclusivity of light is equal toe. 1240 Electron world nanometer derided by and RG not for town. That is actually Kal Toh. Another difference between the liberals that is energy of M six. 54 wide 36 electron world minus energy up and in cold 23 is 13 point 59 Electron world. This gives us relent off immediate for town in this transition equal to 30 no matter.


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