1、ModernQuantumMecSolutionsMaiRevisedEditiJ.J.SakunLate.UniversityofCalifornia,LosBySanFuTmUniversityofHawaii.Mane1234567ContentsFundamentalConceptsQuantumDynamicsTheoryofAngularMomentSymmetryinQuantumMeetApproximationMethodsIdenticalParticlesScatteringTheoryAB,CDhABCD-CDAB-ABCD+ACBD-ACBDACDB+ACDAB=AC

2、,BD-ACD,B+C.ADB-CD.AB2.(a)X=a+Eaaae,tr(X)*2abecausetr(cA)c0.TOC o 1-5 h zolxotr(aX)Htr(aac)=1筆力長x2a(wherevehavtr(au)=trCCo.c+o.c.)=26.)Hencea13丄13o(b)a。*匕(X+%22)訕口包canbeexplicitlyevavithX【乂巧)andi,j=1,2TheresultisX(X2),and勺-仏).oaoaayy+az%azax-laya+ia-axyzdet(c.a)T，|2.Vithoutlossofgenerality,choosenal

3、ongpositivez-q(ia.n*/2)=丄cos/2iasin|/2andifBis(zcosf/2+isinf/2,then ModemQuantumMechanicsSolutions #ModemQuantumMechanicsSolutions4.invariantunderspecifiedhenC甘9a；counter-clockwiseoperation)xyeaxCOS*十a(chǎn)ySfayrotationaboutz-axis(a)Notetr(XY)Ja*IXYla1a11jb|Y|aacythrou；八closureproperty)=a,aEr.Sinceaisadu

4、nnysummationahencetr(XY)=tr(YX).=af|XY*eTherefore(XY)十Y+X+.Takeexpif(A)|a(1+if(A)-丄學(xué))丄283(1+if(a)-I+|a=expifModernQuantModernQuantexpressioninsidesquarebracketisthe(i,j)m(b)|a=|s次X/2M|+jSs*M/2=Z3CHence(1/24.(:GivenA(i=a|iandA|j|jisofform|=|iaeIjwherea,a.arerealnumbers31iJThenorzna+lj).HenceA|中ifAisB

5、ejaclearlyr.h.s.isastatevectordistinctfromconditionthat|iand|jaredegeneratie(ie.a(l/2)(|i+|j)=a|and|*or|i+|jA.7.(a)Letc|afandA|a二a1|a*.ThenproductoveralluigenvalaES,and|?=ModemQuantuaMechanicsModemQuantuaMechanicsModernQuantumMechanics-Solutions(6LetA=Sz，thangf(Sa)(SX/2anll/2(Sz川Thisverifies(aModemQ

6、uantuaMechanicsModemQuantuaMechanicswehave3亠(S/2)/K0-(S-U/2)/M;-卜andaretheprojectionistates8TheorthonormalitypropertyisveobtainS.SjieijkMSkand(SiSj)-(”八“出.Letn=ni+ni+nk,thennsin0cosacosBandXJrNXsinBcosaS+sinsinaS+、xycoapletenesspropertyoftheketspace|.n|b|=1(normalization)Thereforethere(M/2)|.n;+caki

7、ngadvantageofexplicitv_|+卜x+|).Syu豊(T+x_|+|-X+|),S10-Ha(|l(J),|0)and”廠厶（二，帀念ih（+1）12_%八怡（T+，乙血RevriteHasHX%+H22)(|1whereBanalogoustotan*1 #ModernQuantumMechanics-Solutions #ModernQuantumMechanics-Solutions #ModernQuantumMechanics-Solutions #ModernQuantumMechanics-SolutionsTheotherenergyeigenkeccanbe

8、written+ccos(B/2)|+sin(S/iM(+;n)n(cos|+sin|-)(cos|+|+Thefinalmeasurementcorrespondstotheoperat(吟=M(-)M(+;n)M(+)B_2insB-2smeasurementS-H/2beam,whentheS=M/2beamsurvivinz220lizedtounity,isthuscos(8/2)sin(6/2)=(sfinalbeam,setB=r/2,i-ealongOX,andint14a.bntonacompleteorthonormalset.ThisarbitraryAB=0nust16

9、A,BAB十BA-0ThislapliesthacaM|AB(an+a1)0.Ingeneralau4-a,*0ta1aswellasanavhenceitIsnotpossiblofAandB.ThewcrivialMcaseIswhena+a*andsimultaneouselgexiketofAandBwouldappA|a9bva*|a*,bftB|av9bvbf|aftb*(0Hencea*0vorb10,orafbf0genketsarepossiblebutatthecostthatthe(orboth)ofoperatorsAandBarezero17.Uodegeneracy

10、implies|ndefinedbyH)neigenstatewhenIsgiven.Now0vuleA|nisanenergyel&enketButvearegivModernQuancmMechanicModernQuancmMechanic2|Thegeneralizeduncertaintyrelation(1.4.59)is(AAwhereaccordingto(1.4.63)|k+k|AnelentarycalculationleadstoA3AAfAB9hence(1.4.50)veknowthatAA*A*andAB-B-andAModernQuancmMechanicMode

11、rnQuancmMechanic(AAsAB|aA*-X.ChoosenextAt-2Xwhile(AtB|a|lpalsoevidentthatforXioaginary0thei2?therecognitionthat|A|a|(Allltyinthegeneralizeduncertaintyrelation(1.4.59).(c)SinceAxxwemayas/dxnModernQuancmMechanicModernQuancmMechanicHence/dxw6(x,-xw)x,*/dxM6(xf-xM)wherenona6(Fx”)Ischosen.ForApp-wherep-U

12、(|x/dx,and邛|才x(-i)()|M/dxn6(x1-xrt)Usenextexp(2wd2)Jxexp說Elinboveintegi ModernQuantumMechanics-Solutions #ModernQuantumMechanics-Solutions=H2/4and=川/16jcy-F=iM20.Noteexplicit2bothsides+weusesystematicallyorthonormalityconditjTakethenormalizedlinearcombinationandajwThanelementarycalculationsyj222cos0

13、and=2(l-4a2(l-a2)sin：1-心2Maximumforsin2BiswhenB=ir/4,andrhandthemax: ModernQuantumMechanics-Solutions #ModernQuantumMechanics-Solutions=H2/4and=M4/16.2CySS|+二iM+1S=i2/2.Thegenerxy|zforeverifiedfortheequalitycase.;/*B-2soc_一B-2n1sa.1e #ModernQuantumMechanics-Solutions #ModernQuantumMechanics-Solution

14、s%,s】|S/+|,NoteexplicitXszs=M2/4,therefore負；+|(%：hence-0and;+1(AS*)|S：bothsidesofgeneralizedi #ModernQuantumMechanics-Solutions #ModernQuantumMechanics-Solutions #ModernQuantumMechanics-Solutions #ModernQuantumMechanics-Solutionsweusesystematicallyorthonormalitycondit20-Takethenormalizedlinearcombin

15、ationandIa|w1.Thanelementarycalculationsyo2u22oycos3and|(AS)|(l-4a(1-a)sin2Maximumforsin28iswhen6=tt/4,andrh2clearthata=isaminimumandthemaxModernQuantifliMechaiModernQuantifliMechai22.22.ModernQuantifliMechaiModernQuantifliMechai22.22.ZleThisistherigidvailpotstlal(Sne*dien3ionalboxw)2.陀.z(A24)ofAppe

16、ndixAThewavefunctionsaodenergyei;/2/asin(nTx/a),n1.2.3andEn-1:AZ-tbeexcitedstatesNextnotethat222222.pS-*32222wherepandp-K3/SxForrigidwallpotenu扌Cxsinhnxx/aldx2a|yj-xsin(mrx/a)dxa/24.22-f81D()H292/3x2)81n()dx-(2(naoaaasin()0aoaldxa22Thereforetheuncertaintyproductu22扌l(wèi)(m)/6*1);forgroundstatenltforexci

17、tedstateAssumthattheicepickisequivalenttoab&mpointoflengthLtheotherendofvhichisbalancedonafixesnailangle0departureofpickfrovertical,thetorqu一mm.ATaft*、*44whtm ModernQuantumMechanics-Solutions ModernQuantumMechanics-Solutionstomand*Foranyreasonablevaluewege：23.(a)Thecharacteristicequationdet(B-Xl)=0入

18、=bandAbisatwo-folddegenerateelgenv(b)Straightforwardmatrixmultiplicationgiv(ab00、00iabIBAhenceA9B血0-iab0/Theeigenvectors(eigenkets)ofB,togetheouseigenvectorsofAandB.LetX.beelgceigenvectorsareili23uuuwhereBu=入.u.Forthedegenerate入？wehavebu】bua12.ormaltnu.henceu,=0ThereforewechoModernQuantua1ModernQuan

19、tua1ModernQuantua1ModernQuantua1ModernQuantua1ModernQuantua1istvofold-degeneracyverteigenvalue-aofopen24.(a)Therotationaatrixc.f.(3244)actingonawrittenasexp-io.n8/21cos*-ic.nsin?FoiI2x-axisthroughwehave6c-t/2,henceexp-ic(b)Ifvetransformfrombaseketsinrepresentatbasekets,i.e.rotatebyangle-r/2aboutx-ax

20、is,SOjThis(l/2)(l-iux)az(l/2)(l+iox)-|canbeseenbynotingthatif|cisSybasithantransformationisModernQuantua1ModernQuantua1ModernQuantua1ModernQuantua1crf-btEb.：25-Givenisreal.Takeanotherbasisjc,1vc*AjcJ(”心小心|”(dbcb|bb)-.Itisnotnecessarythatandhenceandcasesofproblem24aboveHere|b*匕forSzM|cfs|Sy;+訕山forSyM

21、odernQuantumkModernQuantumk14ModemQuantunMechanics-Solutjwitha9b+Takechegeneralfor|bvitb27.weseethatUcanindeedbeexpress(a)Matrixelementawhere(likevlse)isthea*basistothebfbasis(b)1.Nocechat-(1/(2tM)3SupposeF(r)issphericallysyoaetr:創(chuàng)F(C|L-f14whereqIntegrateoutrilatlonshlpsfD、XfPjUsingtheseriesforaforGoathatlealinductionP(x)-iM3F/3x1.22222(b)(xtpxppx9pp+pxtp),butfrom2iXxp+2iMpx21Xx.pTheclassicalPBforf229x23p23x23p2、2一.XP】cl忘詣喬式w“(2p)-Axpsince12xp,wehave込p2銅iM(x2,p2)clwherealT(t).(b)Notingthata|xtakeexpressiona