量子物理习题

量子物理课堂习题Lecture1:旧量子论1.Hn=2n=3Ce1.9eV,He+Li++,第一、第二波尔轨道半径及电子在这些轨道上的速度电子在基态的结合能第一激发态退激到基态所放光子的波长Lecture2：波粒二象性不确定性原理―,=n12,a2驻波”的说法，束缚在长宽高分别为a，b，c的三维势箱中的粒子（质量为m）的定态能量取值是多少？一原子的激发态发射波长为600nm的光谱线，测得波长的精度为/=107,13aLecture3:波函数薛定谔方程.下列哪些函数不是品优函数，说明理由：=2,,sin,2.试写出下列体系的定态薛定谔方程：（a）He原子（b）H2分子.写出一个被束缚在半径为a的圆周上运动的粒子的Schrodinger方程，并求其解Lecture4:势箱模型(2.12)Fortheparticleinaone-dimensionalboxoflength1,wecouldhaveputthecoordinateoriginatthecenterofthebox.Findthewavefunctionsandenergylevelsforthischoiceoforigin.

2.15AlThJl'oJpiL.iec!「0n-saconju=ateLlmoleculeregardsthesectronsasmovingintheparticle-in-a-boxpotentialofHg+2Lwheretheboxlengthissomewmorethanthelengthoftheconjugatedchain.ThePauliexclusionprinciple(Chapter10)allnomorethantwoelectronstooccupyeachboxlevel.(Thesetwohaveoppositespins.)Forbidiene?CH2=CHCH-CH2ttaketheboxlengthas7.0Aandusethismodeltoestimatewavelengthoflightabsorbedwhenapielectronisexcitedfromthehighest--occupiedtothe1d^anlk：v』,丁&cxpcrimcnlalvalueis217”i.length.(a)Determinetheprobabilityoffindingtheparticleintheleftquarterofthebox.(b)Forwhatvalueofnisthisprobabilityamaximum?(c)Whatisthelimitofthisprobabilityfor?(d)Whatprincipleisillustratedin(c)?(A23)试用一维势箱模型个电子计算如下分子的电子光谱最大吸收波长h3c、■h3c、■H3CHC—C—C-C—N+HHH-l=8A-»CH3CH3（第一吸收峰）。Lecture5:谐振子(4.13)Usetherecursionrelation(4.48)tofindthe=3normalizedharmonic-oscillatorwavefunction.(4.18)Thethree-dimensionalharmonicoscillatorhasthepotential-energyfunction=2+12+1(3.39)Fortheparticleconfinedtoaboxwithdimensionsa,b,andc,findthefollowingvaluesforthestatewithquantumnumbers,,(3.39)Fortheparticleconfinedtoaboxwithdimensionsa,b,andc,findthefollowingvaluesforthestatewithquantumnumbers,,.(a);(b),.Usesymmetryconsiderationsandtheanswertoparta.(c);(d)2.Is2=2?Is=?(3.24)Findtheeigenfunctionsof-2-2/2.Iftheeigenfunctionsaretoremainfinitefor±,whataretheallowedeigenvalues?3.27Evaluatethecommutators(a)[,],(b)[,2],(c)[,],(d)[,(,,)],(e)[,],wheretheHamiltonianoperatorisgivenbyEq.(3.45);(f)[,2]7.4LetAandBbeHermitianoperatorsandletcbearealconstant.(a)ShowthatcAisHermitian.(b)ShowthatA+BisHermitian.7.5(a)Showthat2/2andareHermitian,where(2/2)2/2show222bysolvingtheSchrodingerequation.(b)If==,findthedegreeofdegeneracyofeachofthefourlowestenergylevels.(4.22)Findtheeigenvaluesandeigenfunctionsofforaone-dimensionalsystemwith=for<0,=12for0.2Lecture6:算符与量子力学

that?(d)Showthat一.维势箱(0，(c)Foraone-particlesystem,doesequal+that?(d)Showthat一.维势箱(0，0foraone-particlesystem.兀x兀xV(x)=ASin——cos2——)中的粒子的状态为aa，计算：能量的可能测量值及相应的几率；能量的平均值；求归一化系数A7.37(a)Showthat,foraparticleinaone-dimensionalbox(Fig.2.1)oflength,theabilityofobservingavalueofbetweenand+is—~2-2-11cos(7.111),where1and1.TheconstantN222istobechosensothattheintegralfromminusinfinitytoinfinityof(7.111)isunity.(b)Evaluate(7.111)for=±/2.[Atthesevaluesofthedenominatorof(7.111)iszeroandtheprobabilityreachesalargebutfinitevalue.]Lecture7:角动量(5.17)(a)Showthethreecommutationrelations(5.46)and(5.48)areequivalenttothesinglerelationx=.(b)find[2,](5.24)Calculatethepossibleanglesbetweenandtheaxisfor=2(5.25)Showthatthesphericalharmonicsareeigenfunctionsoftheoperator2+2.(Theproofisshort.)Whataretheeigenvalues?(5.30)Applytheloweringoperatorthreetimesinsuccessionto11(,)andverifythatweobtainfunctionsthatareproportionalto10,11,and0.(7.7)LetbeaHermitianoperator.Showthat2=2andtherefore20.Lecture8：H原子61IfthethreeforceconstantsinProblem4」&areallequal,wehaveathree-dimensionalisotropicharmonicoscillator,(a)Statewhythewavefunctionsforthiscasecanbewrittenas巾=好(b)WhatisthefunctionG?(c)Writeadifferentialequationsatisfiedbyf(r)r(d)UsetheresultsfoundinProblem4.18toshowthattheground-ststewavefunctiondoeshavethefotinf0andverifythattheground-statef(f)satisfiesLhcdifferentialequationin(c).&2TheparticleinasphericalboxhasV=0furrwbandV—coforr>b.ForLhissystems(a)Explainwhy*=机whereR(r)satisfies(6,17),Whatisthefunctionf(出㈤？(b)Solve(6J7)forR[r)forthe1=0statesHints:ThesubslilutionJ?(r)=g(r)/rreduces(6.17)toaneasilysolvedeqiiation.LisetheboiindaiyconditionIhat山iffinitealr=6[seethediscussionafterEq.(6,82)]anduseasecondboundaiyconditiontShowthat*=7¥(sinkr"rforthe/=0statesiwherekm(ImE/h1)11andE=??肥/8m6。withn=1,2,3,…一(Fo「I#t),theenergy-levelformulaismorecomplicated.)64Forasystemoftwononinteract!ngparticlesofmass90X10一，and5.0x10-26ginaone-dimensionalboxoflength1.00x108cm,calculatetheenergiesoithesixloweststationarystates.6A7Fbrthegroundstateofthehydrogcnlikeatom,showthat(r)=3a/2Z,ForIlieHatomgroundstate,findtheprobabilityuffindingiheelectruninLhechs-sicallyforbiddenregion.Lecture9：变分法&1Applythevariationfunction@=e~crtoLhehydrogenatom;choosetheparameterctominimizethevariationalintegral,andcalculatethepercenterrorintheground-stateenergy8.2VerifyEq.(8.13)for⑷冏始83Ifthenormalizedvariationfunction力=(3/户)皿工for0w，宅/i帛appliedtotheparticle-in-a-one-dimensitinal-boitproblem,onefindsthatthevariationalintegralequalszero+whichis痴thanthetrueground-stateenergy.Whatiswrong?8,5(a)Consideracne-pariicle,one-dimensionalsystemwithpotentialenergyV=VQV=0fcr0宏x毛¥and戈w上这，andV=ooelsewhere(where%isaconstant)nPlotVversus尤Usethetrialvariationfunction的=(2/f)l/2sin(irJt/7)for0曰ftQestimatetheground-stalcenergyfor%=淤/用FandcomparewithThetruegroutid-stateenergyE=5-75Q345；fi2//MF-Tbsavetim已inevaluatingintegrals,note阂用加=(明亍|媚+(曲|了|巾Jandexplainwhy胡/随)equalstheparticle-in-a-bexgreund-stateenergy加/gmJ与fb}Forthissystemhusethevariationfunction机~工(‘一工卜TosavelimeTnotethat(:电)isgivenbytheequationafter(S.ll).(Why?)Lecture10微扰论941FortheanharmonicoscillatorwithHamiltonian(9.3)tevaluate炉"forthefirstexcitedstate,takingtheunperturbedsystemastheharmonicoscillator.H=-轰+^kxH=-轰+^kx2+ex3+dx4(93)9.2Considertheune-particle.one-diniensionalsystemwithputential-energy炉=%for1；<a<V=0for。餐上餐Xandymx金，andV=(X)elsewhere,where%="//nF±"Treatthesystemasaperturbedparticleinabox.(a)Findthefirst-orderenergycorrectionforthegeneralstationarystatewithquantumnumbern.(b)Forthegroundstateandforthefirstexcitedstate,compare日⑴+在“withthetrueenergies3.750345炉/川士and20.23604A1/ffl/1.Explainwhy苏母+加丹foreachofthesetwostalesisthesaineasobtainedbythevanationaltreatmeinofProblems8.5aand8,15.见16(a)Foraparticleinasquareboxoflength/withoriginatx=0,y=。,writedownthewavefunctionsandenergylevek(b)Ifthesyslemof(a)isperiiirbedbyH'=bfor#<hw"and*</*#wherebisaconstant,and小=0elsewhere^find必"f<trtheground8taie.FbrthefirstexcitedenergjrJevel,findthe积"valuesandthecorrectieroih-orderwavefunctions.Lecture11：He原子基态9.9ThereismorethanonewaytodivideaHamiltonianHintoanunperturbedpartH0andaperturbationInsteadofthedivision(9.41))and(y.41)Tconsiderthefollowingwayofdividingupthuheliuni-atomHamiltonian;hH'hH'v16门16r2,r12Whataretheunperturbedwavefunctions?Calculate衿")andX"forthegroundstate.(SeeSection9A)Lecture12：自旋与Pauli原理10』CalculatetheanglethatthespinvectorSmakeswiththe?axisforanelectronwithspinfuncriona.10.4⑼ShowthatcommuteswiththeHamillonianforthelithiumatom,(b)Showthat户12and户驾donotcommutewitheachother,(c)Showthat户口andPMcommutewhentheyareappliedtoantisymmetricfunctions.1G+18(a)If(tiespincomponentSfofanelectronismeasured,whatpossiblevaluescanresult?(b)ThefunctionsaandRformacompleteset,soanyonc-dcctronspinfunctioncanbewrittenasalinearcombinationofthem.UseEqs,(10.70)and(10.71)toconstructthetwonormalizedeigenfunctionsofSxwitheigenvalues+；耸and一★市.(c)Supposeameasurementof&foranelectrongivesthevalue+jA;ifameasurementofSxisthencarriedout,givetheprobabilitiesforeachpossibleoutcome,(d)Dothesameasin(b)for&insteadofSx.IntheStern-Gerlachexperiment,abeamofparticles七sentthroughaninhomogeneousmagneticfield,whichsplitsthebeamintoseveralbeamseachhavingparti