量子化学课程习题及标准答案

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?(x,y,z)]?x?(x,y,z)?V?(x,y,z)x?,V?V??0[x

e)

1????????,[x,H]?[x,T?V]?[x,T]?[x2m

f)

??y?z?]?2?y?z?,p?[x?x2x2Chapter04

1：Theone-dimensionalharmonic-oscillatorisatitsfirstexcitedstateanditswavefunctionisgivenas

?1(x)?2(?)3/412xexp(??x)1/42(?)

pleaseevaluatetheexpectationvalues

(averagevalues)ofkineticenergy(T),potentialenergy(V)andthetotalenergy.

Answer:1)Firstofall,checkthenormalizationpropertyofthewavefunction.

2)Evaluatetheexpectationvalueofkineticenergy.

3)Evaluatetheexpectationvalueofpotentialenergy

4)TotalEnergy=T+V

2.Theone-dimensionalharmonic-oscillatorHamiltonianis

?p222??H??2?vmx2mTheraisingandloweringoperatorsforthis

problemaredefinedas

2x??A?1??[p?2?ivmx]x1/2,(2m)??A?Showthat

1??[p?2?ivmx]x1/2(2m),

?A??H??1hvA??2??A??H??1hvA??2?,A?]??hv,[A???]??hvA??,A,[H?andA?areindeedladderShowthatAoperatorsandthattheeigenvaluesarespacedatintervalsofhv.Sinceboththekineticenergyandthepotentialenergyarenonnegative,weexpecttheenergyeigenvaluestobenonnegative.Hencetheremustbeastateofminimumenergy.Operateonthewave

?andthenfunctionforthisstatefirstwithA?andshowthatthelowestenergywithA?]?hvA??,A[H???????eigenvalueis

1hv2.Finally,concludethat,n=0,1,2,…

1E?(n?)hv2Answer:

1）Writedownthedefinitionofoperator

d?x??i?pdx

2)Expandtheoperatorsinfullform.

?d222?H???2?vmx22mdx1dA??[?i??2?vmxi]dx2m1dA??[?i??2?vmxi]dx2m3)Evaluatethecorrespondingcombinationof

operators

22

1d1dA?A??[?i??2?vmxi][?i??2?vmxi]dxdx2m2m1ddd?[?i?[?i??2?vmxi]?2?vmxi[?i??2?vmxi]]2mdxdxdx21ddd22222?[???2?vm??2?vm?x?2?vmx??4?vmx]22mdxdxdx21d22222?[???2?vm??4?vmx]22mdx?2d211222????2?vmx?hv?H?hv22mdx221d1dA?A??[?i??2?vmxi][?i??2?vmxi]dxdx2m2m1ddd?[?i?[?i??2?vmxi]?2?vmxi[?i??2?vmxi]]2mdxdxdx21dd2d2222?[???2?vm??2?vm?x?2?vmx??4?vmx]22mdxdxdx22?d11222????2?vmx?hv?H?hv22mdx2222?d1d222?HA??[??2?vmx][?i??2?vmxi]22mdxdx2m221?ddd222?[?[?i??2?vmxi]?2?vmx[?i??2?vmxi]]2dxdx2m2mdx3321i?ddd2d222233?[?2?v?i??v?ix?2?vmxi??4?v32dxdxdx2m2mdx221d?d222??A?H[?i??2?vmxi][??2?vmx]2dx2mdx2m21i?3d3dd2222223?[?4?vmxi??2?vmxi???v?ix?4?v32dxdx2m2mdx11d2d22[?2?v?i?4?vmxi?]?hv[?i?]?dx2m2mdx11d1hv[2?vmxi]?hv[?i??2?vmxi]?hvA?2m2mdx2m?A?AH??H??

Inthesamemanner,wecanget

?A?AH??hvAH???4)SubstitutingtheabovecommunicatorsintotheSchroeidngerequation,weget

???E?H?A??[AH??hvA]??AE??hvA??(E?hv)A?H???????A??[AH??hvA]??AE??hvA??(E?hv)A?H??????

?andA?areindeedladderThisshowsthatAoperatorsandthattheeigenvaluesarespacedatintervalsofhv.

5)Supposethat?istheeigenfunctionwiththelowesteigenvalue.

?????E?Hlowest

AccordingtothedefinitionofA_operator,we

have

As?istheeigenfunctionwiththelowesteigenvalue,theaboveequationisfulfilledifandonlyif

?A??(E?hv)A?H??A???0

Operatingonthewavefunctionforthisstate?andthenwithA?leadstofirstwithA??11??hv]??H???hv?A?A???0?[H22

Therefore,thelowestenergyis1/2hv.

1?H??(n?hv)?,2n?0,1,2,3?

Chapter05

1.Forthegroundstateoftheone-dimensionalharmonicoscillator,computethestandarddeviations?xand?pxandcheckthattheuncertaintyprincipleisobeyed.Answer:

1)Thegroundstatewavefunctionoftheone-dimensionalharmonicoscillatorisgivenby

??(?)?e?14141??x22

2)Thestandarddeviations?xand?pxaredefinedas

?x?x22?x

2(?p)2?(?p)2??p2

Theproductof?xand?pisgivenby

?x?p?1????2?22???42

2Itshowsthattheuncertaintyprincipleis

obeyed.

2.(a)Showthatthethreecommutationrelations[L?,L?]=i?L?,[L?,L?]=i?L?,[L?,L?]=i?L?

??L??i?L?(b)areequivalenttothesinglerelationL?]?,LFind[LAnswer:1):

xyzyzxzxy2xy?????Li?Lj?LkLxyz????????L??(Li?Lj?Lk)?(Li?Lj?Lk)Lxyzxyz???????LxLyk?LxLzj?LyLxk?LyLzi?LzLxj?LzLyi????(LyLz?LzLy)i?(LzLx?LxLz)j?(LxLy?LyLx)k???????[Ly,Lz]i?[Lz,Lx]j?[Lx,Ly]k?i?(Lxi?Lyj?Lzk)[Lz,Lx]?i?Ly[Lx,Ly]?i?Lz?[Ly,Lz]?i?Lx

2):

[L2x,Ly]?Lx[Lx,Ly]?[Lx,Ly]Lx?Lx(i?Lz)?(i?Lz)Lx?i?(LxLz?LzLx)

3.CalculatethepossibleanglesbetweenLandthezaxisforl=2.

Answer:

ThepossibleanglesbetweenLandthezaxisareequivalenttheanglesbetweenLandLz.Hence,theanglesaregivenby:

L?2(2?1)??6?mzCos??L???

???35.26,65.91,90.00,114.10,144.7

4.

Complete

this

equation:

3m33m?LzYl?m?Yl

Chapter06

1.Explainwhyeachofthefollowingintegralsmustbezero,wherethefunctionsarehydrogenlikewavefunctions:(a);(b)Answer:

Both3p-1and3p0areeigenfunctionsofLz,witheigenvaluesof-1and0,respectively.Therefore,theaboveintegralscanbesimplifiedas

a)duetoorthogonalizationpropertiesofeigenfunctions?3p??12p|3p?02pLzz1z?11?1b)0

2.Useparitytofindwhichofthefollowingintegralsmustbezero:(a);(b)

2

;(c).Thefunctionsintheseintegralsarehydrogenlikewavefunctions.Answer:

1)b)andc)mustbezero.

3.Forahydrogenatominapstate,thepossibleoutcomesofameasurementofLzare–?,0,and?.Foreachofthefollowingwavefunctions,givetheprobabilitiesofeachofthesethreeresults:(a)?;(b)?;(c)?.Thenfindforeachofthesethreewavefunctions.

Answer:

a)?2p??2p,therefore,theprobabilitiesare:0%,100%,0%

2pz2py2p1z0?2px1?(?2p1??2p?1)2,theprobabilitiesare

50%,0%,50%.

?2p,theprobabilitiesare100%,0%,0%b)0，0，1

1/2

4.Ameasurementyields2?forthemagnitudeofaparticle’sorbitalangularmomentum.IfLxisnowmeasured,whatarethepossibleoutcomes?

1

Answer:

1):Sincethewavefunctionisthe

2

eigenfunctionofL,ameasurementofthemagnitudeoftheorbitalangularmomentumshouldbe

L(L?1)??2??L,

ThepossibleoutcomeswhenmeasureLxare-1,0,1

?1Chapter07

1.Whichofthefollowingoperatorsare

2222

Hermitian:d/dx,i(d/dx),4d/dx,i(d/dx)?Answer：

Anoperatorinone-DspaceisHermitianif

*???A?dx??(A?)dx??

*a)

d?*??dxdx???d*????(?)dxdx*???d?d????dx????dxdxdx**

b)

d?*??idxdx?i??d*???(i?)dxdx*?d?d??i??dx??i????dxd*c)

d?*d??4dx?4??dx2dx**2???d?d??4?dxdx*?*d?d?d???4?dx??4?dxdxdxd???4?dx2dx2*??d??4?d2

Thisoperatorcanbewrittenasaproductof1Dkineticoperatorandaconstant.Hence,it’sHermitian.

d)AsthethirdoperatorisHermitian,thisoperatorisnotHermitian.

?andB?areHermitianoperators,prove2.IfA?B?isHermitianifandonlythattheirproductA?andB?andB?commute.(b)IfA?areifA?B?)isHermitian.?+B?AHermitian,provethat1/2(A?Hermitian?(d)Is1/2(x?p?+p?x?)(c)Isx?pxxxHermitian?Answer:1)

IfoperatorAandBcommute,wehave

?B??A?B??0???B?A??B?AA*??????[(AB?BA)?]d??0

??B?)?]*d??0??B?A???[(A**???????[AB?]d????[BA?]d?

OperatorAandBareHermitian,wehave

??]*d??(A??)*(B?B?A??)d???*A??d????[B??Therefore,whenAandBcommute,the

followingequationfulfills.Namely,ABisalsoHermitian.

**??????[AB?]d????AB?d?

2)

1????1*??*???[(AB?BA)]?d??[?AB?d???BA?d?]?2??2*OperatorAandBareHermitian,weget

11*??*????)*d??A[??AB?d????BA?d?]?[??(B221????*?????(AB?)d????[(BA?AB)?]*d?21????*1??*?????(AB?BA)?d????[(AB?BA)?]d?22

Theaboveequationshowsthattheoperator1/2[AB+BA]isHermitian.

c)xpxisnotHermitiansincebothxandpxareHermitiananddonotcommute.d)Yes

Chapter08

1.Applythevariationfunction??e?crtothehydrogenatom;choosetheparameterctominimizethevariationalintegral,andcalculatethepercenterrorintheground-stateenergy.Solution：

1）Therequirementofthevariationfunctionbeingawell-behavedfunctionrequiresthatcmustbeapositivenumber.

2）checkthenormalizationofthevariationfunction.

???d???e??d??H?**?2cr2rdr?Sin(?)d??d???c

333)Thevariationintegralequalsto

121c12*w???(???)?d???(????*?2r?2??d??*c31?2??cr23?cr3?(?)?d???2c?e[(2?)e]rdr?4c??r?rr?r1?c(c?2)2*c32

4)Theminimumofthevariationintegralis

?w1?c?1?0?c?1?w???c2

5)Thepercenterrorinthegroundstateis0%

2.Ifthenormalizedvariationfunction??(3/l)xfor0≤x≤lisappliedtotheparticle-in-a-one-dimensional-boxproblem,onefindsthatthevariationintegralequalszero,whichislessthanthetrueground-stateenergy.Whatiswrong?Solution:

Thecorrecttrailvariationfunctionmustbesubjecttothesameboundaryconditionofthegivenproblem.Fortheparticleina1Dboxproblem,thecorrectwavefunctionmustequaltozeroatx=0andx=l.However,thetrialvariationfunction??(3/l)xdoesnotfulfilltheserequirement.Thevariationintegralbased

31/231/2onthisincorrectvariationfunctiondoesnotmakeanysense.

3.Applicationofthevariationfunction(wherecisavariationparameter)to

aproblemwithV=af(x),whereaisapositiveconstantandf(x)isacertainfunctionofx,givesthevariationintegralasW=c?2/2m

3

+15a/64c.FindtheminimumvalueofWforthisvariationfunction.Solution:

??e?cx2c?a5d(?15)()3a3?w2m64c2??0?c??1?cdc42m??wmin5()am?1332??0.72598a4m4?231414343221414

4.In1971apaperwaspublishedthatappliedthenormalizedvariationfunction

Nexp(-br2a02-cr/a0)tothehydrogenatomandstatedthatminimizationofthevariationintegralwithrespecttotheparametersbandcyieldedanenergy0.7%abovethetrueground-stateenergyforinfinitenuclearmass.Withoutdoinganycalculations,statewhythisresultmustbewrong.Solution:

Fromtheevaluationofexercise1,weknowthatthevariationfunctionexp(-cr)givesnoerrorinthegroundstateofhydrogenatom.ThisfunctionisaspecialcaseofthenormalizedvariationfunctionNexp(-br2a02-cr/a0)whenbequalstozero.Therefore,adoptingthenormalizedvariationfunctionasatrialvariationfunctionshouldalsohavenoerrorinthegroundstateenergyforhydrogenatom.

5.Provethat,forasystemwitha

??d??E,if?isnondegenerategroundstate,??Hanynormalized,well-behavedfunctionthatisnotequaltothetrueground-statewave

*0function.(E0isthelowest-energyeigenvalue

?)ofHSolution:

AstheeigenfunctionsoftheHermitianoperatorHformacompleteset,anywell-behavedfunctionwhichissubjecttothesameboundaryconditioncanbeexpandedasalinearcombinationoftheeigenfunctionoftheHermitianoperator,namely,

???ci?i,where?sareeigenfunctionsofi

i?0?HermitianoperatorH,cisareconstant.

Theexpectationvalueof???withrespecttotheHermitianoperatoris

*????*?*?*???H?d???(?ci?i)H?cj?jd???ci?cj??iH?i?0j?0i?0j?02??ci?02?*i?cE???ccE??cjjij*iiij?0i?0i?0?2?2i?1i?0???2iEi?c0E0??cii?1??c0E0??ciE0?E0?ci?E0

Chapter09,10

1.FortheanharmonicoscillatorwithHamiltonian

?2d1234?H???kx?cx?dx2mdx22,evaluateE(1)

forthefirstexcitedstate,takingtheunperturbedsystemastheharmonicoscillator.Solution:

Thewavefunctionofthefirstexcitedstateoftheharmonicoscillatoris

?1?(4?314?)xe??x22

Hence,thefirstordercorrecttoenergyofthefirstexcitedstateisgivenby

4?'???H?1dx??()xe*1314??x22?(c?x?d?x)(4?3344?3?)x14?(4?3?)12?xe2??x2d?xdx?d(4?)12?xe6??x215ddx?4?

2.Considertheone-particle,one-dimensionalsystemwithpotential-energyV=V0for

13l?x?l44,V=0for

0?x?1l4and

3l?x?l4

22andV=∞elsewhere,whereV0=?/ml.Treatthesystemasaperturb