M40_KNIG9404_ISM_C40.pdf

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1、40ONE-DIMENSIONALQUANTUMMECHANICSConceptualQuestions40.1.Asthewavefunctionofthequantumsystemoscillates,thequantumprobabilitydensity(thesquareofthewavefunction)alsooscillates.Theaveragebehaviorofthequantumsystemisthentheaverageoftheprobabilitydensityofthequantumsystem.40.2.Five.40

2、.3.n=6becausetherearesixpeaks.=40.4.ηabc=>ηηη.isindependentofL.η=.2(mU0−E)==ηηab==ηc=2(10eV5eV)mm−−2(16eV10eV)40.5.(a)Thespacingbetweenthenodesincreasesasxincreases.Thishappensbecausetheparticleslowsdownasitapproachestheturningpoints.Withslowerspeed,theparticle’sdeBrogliewaveleng

3、thλ=hmv/increases.ThewavefunctionisastandingdeBrogliewave,soincreasedwavelengthmeansincreasedspacebetweenthenodes.(b)Theheightsoftheantinodesincreaseasxincreases.Thisisbecausetheparticle,duetoitsslowerspeed,spendsmoretimeatlargerxandisthusmorelikelytobefoundatlargerx.Ahigherproba

4、bilityforfindingtheparticleatlargerximplieslargervaluesforψ()xandthushigherantinodes.(c)Theaboveconsiderationsleadtothegraphofthen=8wavefunctionshownbelow.Thewavefunctionhaseightantinodes.©Copyright2013PearsonEducation,Inc.Allrightsreserved.Thismaterialisprotectedunderallcopyrigh

5、tlawsastheycurrentlyexist.Noportionofthismaterialmaybereproduced,inanyformorbyanymeans,withoutpermissioninwritingfromthepublisher.40-140-2Chapter4040.6.Wavefunctionaisanantibondingorbitalandwavefunctionbisabondingorbital.Youcantellbecauseinthebottomonetheelectronhasahighprobabili

6、tyofbeingfoundbetweenadjacentprotons,indicatingthattheadjacentprotonsaresharingtheelectron.40.7.()()()()PPPPtunneld>>>tunnelatunnelbtunnelc.Thetunnelingprobabilitydecreasesifeitherthepotentialbarrieris−2w=madewideroriftheenergyisloweredrelativetothetopofthebarrier.BecausePe=ηandη

7、=,tunnel2(mU0−E)changingthebarrierwidthbyafactorof2changestheexponentbyafactorof2whereaschangingUE0−byafactorof2changestheexponentbyafactorofonly2.Thusbarrierchasalarger(negative)exponentthanbarrierb,bothbeinglargerthanbarriera,leadingto()PPPtunnelc<<()()tunnelbtunnela.Relativeto

8、barriera,thewidthofbarrierdisnarrowerbya