量子化学课程习题及标准答案Word格式文档下载.docx
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量子化学课程习题及标准答案Word格式文档下载.docx
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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,estimatetheprobabilitythattheresultwillliebetween2.000Å
and2.001Å
.
whent=0,thewavefunctionissimplifiedas
Chapter02
1.Consideranelectroninaone-dimensionalboxoflength2.000Å
withtheleftendoftheboxatx=0.(a)Supposewehaveonemillionofthesesystems,eachinthen=1state,andwemeasurethexcoordinateoftheelectronineachsystem.Abouthowmanytimeswilltheelectronbefoundbetween0.600Å
and0.601Å
?
Considertheintervaltobeinfinitesimal.Hint:
Checkwhetheryourcalculatorissettodegreesorradians.(b)Supposewehavealargenumberofthesesystems,eachinthen=1state,andwemeasurethexcoordinateoftheelectronineachsystemandfindtheelectronbetween0.700Å
and0.701Å
in126ofthemeasurements.Inabouthowmanymeasurementswilltheelectronbefoundbetween1.000Å
and1.001Å
Solution:
a)Ina1Dbox,theenergyandwave-functionofamicro-systemaregivenby
therefore,theprobabilitydensityoffindingtheelectronbetween0.600and0.601Å
is
b)Fromthedefinitionofprobability,theprobabilityoffindinganelectronbetweenxandx+dxisgivenby
Asthenumberofmeasurementsoffindingtheelectronbetween0.700and0.701Å
isknown,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.
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
Fortheparticleinacertainrectangularwell,theEfulfillwith
SubstitutingintotheVandE,weget
Chapter03
1.If
f(x)=3x2f(x)+2xdf/dx,giveanexpressionfor
Extractingf(x)fromtheknownequationleadstotheexpressionofA
2.(a)Showthat(
+
)2=(
)2foranytwooperators.(b)Underwhatconditionsis(
)2equalto
2+2
2?
a)
IfandonlyifAandBcommute,(
)2equalsto
2
3.If
=d2/dx2and
=x2,find(a)
x3;
(b)
(c)
f(x);
(d)
f(x)
c)
d)
4.Classifytheseoperatorsaslinearornonlinear:
(a)3x2d2/dx2;
(b)()2;
(c)∫dx;
(d)exp;
(e)
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.
a)Lisalinearoperator
c)
6.Wedefinethetranslationoperator
by
f(x)=f(x+h).(a)Is
alinearoperator?
(b)Evaluate(
)x2.
a)Thetranslationoperato
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