solutions to problems in sakurai's quantum mechanics - p. saltsidis, b. brinne

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1、SolutionstoProblemsinQuantumMechanicsP.Saltsidis,additionsbyB.Brinne1995,19990MostoftheproblemspresentedherearetakenfromthebookSakurai,J.J.,ModernQuantumMechanics,Reading,MA:Addison-Wesley,1985.ContentsIProblems31FundamentalConcepts......................52Quantum

2、Dynamics........................73TheoryofAngularMomentum..................144SymmetryinQuantumMechanics................175ApproximationMethods.....................19IISolutions231FundamentalConcepts......................252QuantumDynamics........................

3、363TheoryofAngularMomentum..................754SymmetryinQuantumMechanics................945ApproximationMethods.....................10312CONTENTSPartIProblems31.FUNDAMENTALCONCEPTS51FundamentalConcepts01.1ConsideraketspacespannedbytheeigenketsfjaigofaHer-mitiano

4、peratorA.Thereisnodegeneracy.(a)ProvethatY0(A;a)0aisanulloperator.(b)Whatisthesignicanceof00Y(A;a)?000a;a000a6=a1(c)Illustrate(a)and(b)usingAsetequaltoSofaspinsystem.z21~1.2AspinsystemisknowntobeinaneigenstateofS
n^with2eigenvalueh=2,wheren^isaunitvectorlyingin

5、thexz-planethatmakesananglewiththepositivez-axis.(a)SupposeSismeasured.Whatistheprobabilityofgetingx+h=2?(b)EvaluatethedispersioninS,thatis,x2h(S;hSi)i:xx(Foryourownpeaceofmindcheckyouranswersforthespecialcases=0,=2,and.)1.3(a)ThesimplestwaytoderivetheSchwarzi

6、nequalitygoesasfollows.Firstobserve(hj+hj)
(ji+ji)0foranycomplexnumber;thenchooseinsuchawaythattheprecedinginequalityreducestotheSchwarzinequility.6(b)Showthattheequilitysigninthegeneralizeduncertaintyre-lationholdsifthestateinquestionsatises
Aji=
Bjiwith

7、purelyimaginary.(c)ExplicitcalculationsusingtheusualrulesofwavemechanicsshowthatthewavefunctionforaGaussianwavepacketgivenby"#200ihpix(x;hxi)02;1=4hxji=(2d)exp;2h4dsatisestheuncertaintyrelationqqh22h(
x)ih(
p)i=:2Provethattherequirement00hxj
xji=(imaginarynu

8、mber)hxj
pjiisindeedsatisedforsuchaGaussianwavepacket,inagreementwith(b).1.4(a)Letxandpbethecoordinateandlinearmomentuminxonedimension.EvaluatetheclassicalPoi