HOMEWORK#7PROBLEMSET DUE:Wednesday,November16th1.Supposethatyouhaveanelectroninaone-dimensionalinfinite(square)wellanditcouldbemeasuredwhileinitsgroundstate.(a.)Whatwouldbetheprobabilityoffindingitsomewhereintheregion0<x<L/4?(b.)Whatwouldbetheprobabilityoffindingitinaverynarrowregion∆x=0.01*L-wide,centeredatx=5L/8?Assumethewellisx=0toL.2.(a.)Find<p>and<p2>fortheground-statewavefunctionoftheinfinitesquarewellpotential.(b.)Whatdoyouthinktheresultsshouldhavebeen,usinglogicalone,plainandsimple?3.Abeamofelectrons,eachwithenergyE=0.1*V0,areincidentonapotentialstepwithV0=2eV.(Thisisoftheorderofmagnitudeoftheworkfunctionforelectronsatthesurfaceofmetals.)Graphtherelativeprobability|Ψ|^2ofparticlespenetratingthestepuptoadistancex=1nm(or,roughlyfiveatomicdiameters).4.NormalizethewavefunctionforthesecondexcitedstateofthequantumSHO(simpleharmonicoscillator).Notethatthesecondexcitedstateisn=2,sincen=0isthegroundstateinthiscase.5.Determinetheexpectationvalueofp2foraparticleinaninfinitesquarewellforthethirdexcitedstate.(a.)Usetheoperatormethodand(b.)theenergyequation.Note,thatthethirdexcitedstatemeansn=4(becausen=1isgroundstate;n=0meansnowavefunction).1.(a.)Thewavefunctionforthen=1level,thegroundstate,isgivenbytheequationfromclassasΨ1(x)=√(2/L)*sin(π*x/L)

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Youareofcoursealsoallowedtodopartb.thelongway,integratingover(5L/8)±0.01*L2.(b.)Theparticleisequallylikelytobemovinginthe–xasinthe+xdirection,soitsAVERAGEmomentumiszero;BUT,itsaveragep^2shouldberelatedtoE,sinceE=p^2/2m(KE,assumingnopotential)as2*m*EandweknewEalreadyfromclassandtextexamples.3.Wecanwritethewavefunctionforaparticleontheinsideofabarrierasarealexponentialaslearnedduringlecture.IfwecalltheregionbeforethestepatleftIandtheregionwithinthestepatrightIIthenΨII(x)=C*e–kII*x,wherekII=√[2*m*(V0–E)]/h-bar.(WechoseonlythenegativeexponentialversionnotpositiveorsumincludingpositivetermsothatΨII->0asx->∞.)ThismakesΨIIadecreasingexponentialtowardstheright.So,theparticledensityinregionIIisproportionalto|ΨII|^2=|C|^2*e–2*kII*xsonowwejustneedtosolveforCusingboundaryconditions,continuity,smoothness,andnormalization.Alternatively,wecanrecognizethatthisequationalreadygetsusthegeneralshape.Thepicturefollowsonthenextpage.ItisNOTnecessarytogetthex-andy-valuesexactlyright,justthecorrectshape,plusthetransmissioncoefficientforoutto1nm,eitherindicatedintheplotorwrittendownseparately.Now,fortransmissionprobabilityweconsiderT=1/[1+V0^2*sinh^2(kII*L)/(4*E*(V0–E))]

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kII=√[2*(9.1x10^-31kg)*(3.2x10^-19–3.2x10^-20J)]/[6.6x10^-34J-s/(2*π)]=>T=1/[1+(2eV)^2*sinh^2(6.87x10^9m^-1*10^-9m)/(4*2eV*(2-0.2eV))]=1.5x10-5FullcreditalsoforusinganapproximationforTforacaseofE<<V0(oppositeofclass)4.Y2=A*(2*a*x^2–1)*exp(-a*x^2/2),usingthesecondHermitepolynomial.Theintegralfrom–∞to+∞ofY2^2mustbe1.0Integralfrom–∞to+∞ofA^2*(2*a*x^2–1)^2*exp(-a*x^2)orA^2*[4*a^2*x^4*exp(-a*x^2)–4*a*x^2*exp(-a*x^2)+exp(-a*x^2)].Usingtheintegraltables(morespecificallybothequationA6.2andpageA-11,Appendix6)it’sA^2*[8*a^2*3/(8*a^2)*√(π/a)–4*a*(2/4a)*√(π/a)+(2/2)*√(π/a)]=(3–2+1)*√(π/a)*A^2So,A^2*2*√(π/a)=1andA^2=√(a/π)/2=>A=(a/π)^(1/4)*(1/√2),whichisperfectlyconsistentwithp.224inChapter6,topright.5.(a.)Repeattheprocedurefrompages(21)through(23)oftheChapter6LectureNoteswhichIemailedoutthisweek,exceptnowforn=4,notn=2.(b.)E_n=n^2*π^2*h-bar^2/(2*m*L^2)=n^2*h^2/(8*m*L^2)=>E_4=8*π^2*h-bar^2/(m*L^2)OR2*h^2/(m*L^2)