量子化学课程习题及标准答案.docx

量子化学课程习题及标准答案.docx

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量子化学课程习题及标准答案

量子化学习题及标准答案

Chapter01

1.Acertainone-particle,one-dimensionalsystemhas

whereaandbareconstantsandmistheparticle’smass.Findthepotential-energyfunctionVforthissystem.（Hint:

Usethetime-dependentSchrodingerequation.）

Solution：

Asψ（x,t）isknown,wecanderivethecorrespondingderivatives.

Accordingtotime-dependentSchroedingerequation,

substitutingintothederivatives,weget

2.Atacertaininstantoftime,aone-particle,one-dimensionalsystemhas

whereb=3.000nm.Ifameasurementofxismadeatthistimeinthesystem,findtheprobabilitythattheresult（a）liesbetween0.9000nmand0.9001nm（treatthisintervalasinfinitesimal）;（b）liesbetween0and2nm（usethetableofintegrals,ifnecessary）.（c）Forwhatvalueofxistheprobabilitydensityaminimum?

（Thereisnoneedtousecalculustoanswerthis.）（d）Verifythat

isnormalized.

Solution：

a）Theprobabilityoffindinganparticleinaspacebetweenxandx+dxisgivenby

b）

c）Clearly,theminimumofprobabilitydensityisatx=0,wheretheprobabilitydensityvanishes.

d）

3.Aone-particle,one-dimensionalsystemhasthestatefunction

whereaisaconstantandc=2.000Å.Iftheparticle’spositionismeasuredatt=0,estimatetheprobabilitythattheresultwillliebetween2.000Åand2.001Å.

Solution：

whent=0,thewavefunctionissimplifiedas

Chapter02

1.Consideranelectroninaone-dimensionalboxoflength2.000Åwiththeleftendoftheboxatx=0.（a）Supposewehaveonemillionofthesesystems,eachinthen=1state,andwemeasurethexcoordinateoftheelectronineachsystem.Abouthowmanytimeswilltheelectronbefoundbetween0.600Åand0.601Å?

Considertheintervaltobeinfinitesimal.Hint:

Checkwhetheryourcalculatorissettodegreesorradians.（b）Supposewehavealargenumberofthesesystems,eachinthen=1state,andwemeasurethexcoordinateoftheelectronineachsystemandfindtheelectronbetween0.700Åand0.701Åin126ofthemeasurements.Inabouthowmanymeasurementswilltheelectronbefoundbetween1.000Åand1.001Å?

Solution:

a）Ina1Dbox,theenergyandwave-functionofamicro-systemaregivenby

therefore,theprobabilitydensityoffindingtheelectronbetween0.600and0.601Åis

b）Fromthedefinitionofprobability,theprobabilityoffindinganelectronbetweenxandx+dxisgivenby

Asthenumberofmeasurementsoffindingtheelectronbetween0.700and0.701Åisknown,thenumberofsystemis

2.Whenaparticleofmass9.1*10-28ginacertainone-dimensionalboxgoesfromthen=5leveltothen=2level,itemitsaphotonoffrequency6.0*1014s-1.Findthelengthofthebox.

Solution.

3.Anelectroninastationarystateofaone-dimensionalboxoflength0.300nmemitsaphotonoffrequency5.05*1015s-1.Findtheinitialandfinalquantumnumbersforthistransition.

Solution:

4.Fortheparticleinaone-dimensionalboxoflengthl,wecouldhaveputthecoordinateoriginatthecenterofthebox.Findthewavefunctionsandenergylevelsforthischoiceoforigin.

Solution:

Thewavefunctionforaparticleinaone-dimernsionalboxcanbewrittenas

Ifthecoordinateoriginisdefinedatthecenterofthebox,theboundaryconditionsaregivenas

CombiningEq1withEq2,weget

Eq3leadstoA=0,or

=0.Wewilldiscussbothsituationsinthefollowingsection.

IfA=0,Bmustbenon-zeronumberotherwisethewavefunctionvanishes.

IfA≠0

5.Foranelectroninacertainrectangularwellwithadepthof20.0eV,thelowestenergylies3.00eVabovethebottomofthewell.Findthewidthofthiswell.Hint:

Usetanθ=sinθ/cosθ

Solution:

Fortheparticleinacertainrectangularwell,theEfulfillwith

SubstitutingintotheVandE,weget

Chapter03

1.If

f（x）=3x2f（x）+2xdf/dx,giveanexpressionfor

.

Solution：

Extractingf（x）fromtheknownequationleadstotheexpressionofA

2.（a）Showthat（

+

）2=（

+

）2foranytwooperators.（b）Underwhatconditionsis（

+

）2equalto

2+2

+

2?

Solution:

a）

b）

IfandonlyifAandBcommute,（

+

）2equalsto

2+2

+

2

3.If

=d2/dx2and

=x2,find（a）

x3;（b）

x3;（c）

f（x）;（d）

f（x）

Solution:

a）

b）

c）

d）

4.Classifytheseoperatorsaslinearornonlinear:

（a）3x2d2/dx2;（b）（）2;（c）∫dx;（d）exp;（e）

.

Solution:

Linearoperatorissubjecttothefollowingcondition.

a）Linear

b）Nonlinear

c）Linear

d）Nonlinear

e）Linear

5.TheLaplacetransformoperator

isdefinedby

（a）Is

linear?

（b）Evaluate

（1）.（c）Evaluate

eax,assumingthatp>a.

Solution:

a）Lisalinearoperator

b）

c）

6.Wedefinethetranslationoperator

by

f（x）=f（x+h）.（a）Is

alinearoperator?

（b）Evaluate（

）x2.

Solution:

a）Thetranslationoperato