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1、量子化学习题及标准答案Chapter011.Acertainone-particle,one-dimensionalsystemhas,whereaandbareconstantsandmistheparticle’smass.Findthepotential-energyfunctionVforthissystem.(Hint:Usethetime-dependentSchrodingerequation.)Solution：AsY(x,t)isknown,wecanderivethecorre
2、spondingderivatives.Accordingtotime-dependentSchroedingerequation,substitutingintothederivatives,weget2.Atacertaininstantoftime,aone-particle,one-dimensionalsystemhas,whereb=3.000nm.Ifameasurementofxismadeatthistimeinthesystem,findtheprobabilitythatth
3、eresult(a)liesbetween0.9000nmand0.9001nm(treatthisintervalasinfinitesimal);(b)liesbetween0and2nm(usethetableofintegrals,ifnecessary).(c)Forwhatvalueofxistheprobabilitydensityaminimum?(Thereisnoneedtousecalculustoanswerthis.)(d)Verifythatisnormalized.S
4、olution：a)Theprobabilityoffindinganparticleinaspacebetweenxandx+dxisgivenbyb)c)Clearly,theminimumofprobabilitydensityisatx=0,wheretheprobabilitydensityvanishes.d)3.Aone-particle,one-dimensionalsystemhasthestatefunctionwhereaisaconstantandc=2.000Å.Ifth
5、eparticle’spositionismeasuredatt=0,estimatetheprobabilitythattheresultwillliebetween2.000Åand2.001Å.Solution：whent=0,thewavefunctionissimplifiedasChapter021.Consideranelectroninaone-dimensionalboxoflength2.000Åwiththeleftendoftheboxatx=0.(a)Supposeweh
6、aveonemillionofthesesystems,eachinthen=1state,andwemeasurethexcoordinateoftheelectronineachsystem.Abouthowmanytimeswilltheelectronbefoundbetween0.600Åand0.601Å?Considertheintervaltobeinfinitesimal.Hint:Checkwhetheryourcalculatorissettodegreesorradians
7、.(b)Supposewehavealargenumberofthesesystems,eachinthen=1state,andwemeasurethexcoordinateoftheelectronineachsystemandfindtheelectronbetween0.700Åand0.701Åin126ofthemeasurements.Inabouthowmanymeasurementswilltheelectronbefoundbetween1.000Åand1.001Å?Solu
8、tion:a)Ina1Dbox,theenergyandwave-functionofamicro-systemaregivenbytherefore,theprobabilitydensityoffindingtheelectronbetween0.600and0.601Åisb)Fromthedefinitionofprobability,theprobabilityoffindinganelectronbetweenxandx+dxisgivenbyAsthenumberof
