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Contents1Assumptions11.1Stationary-stateDiracequation.....................11.2Paulimatrices...............................11.3Totalangularmomentum.........................31.4Spin-angularfunctions..........................31.5Commutationrules............................41.5.1...................................41.5.2...................................41.5.3...................................51.5.4...................................51.5.5...................................61.5.6...................................61.6Otherformulas..............................71.6.1...................................71.6.2...................................81.6.3...................................81.6.4...................................101.6.5...................................101.6.6...................................112Lemmas122.1Lemma1..................................122.2Lemma2..................................143Completesetofcommutingobservables163.1Thesetofcommutingobservables....................163.2Themutualeigenstates..........................194SolvingtheDiracequation214.1TheHamiltonian.............................214.2Theradialwavefunction.........................234.3Energylevel................................265Non-relativisticlimit316Thewavefunction33Chapter1AssumptionsIndoingtheDirachydrogenatomproblem,wemustbeawareofwhichequationsorresultswecaninfactuse,whilemanyoftheresultswecannot.Letusenumerateallwhatwehavealreadyknownonthisproblem,astheassumptionsforthisproblem.1.1Stationary-stateDiracequationˆHψ=cˆ~α·ˆ~p+mc2ˆβ−e21rψ=Eψ(1.1)Formatricesweaddahat,whilefor3DEuclideanvectorsweaddanarrow.Thee1istheeffectivechargeasiseasiertouseinSIunits:e1=e√4π0(1.2)Inaddition,wedenotemastherestmassforanelectron(∼9.11E−31kg),anditwillneverdenotetherelativisticmassinthecontext.1.2Paulimatricesˆσx=0110;ˆσy=0−ii0;ˆσz=100−1(1.3)ˆ~s=~2ˆ~σ(1.4)1ThepropertiesofPaulimatricesaredemonstratedbelow:ˆσ2x=ˆσ2y=ˆσ2z=1(1.5){ˆσi,ˆσj}=2δij(1.6)Here{···}istheanti-commutationoperatorandδhereistheKroneckerdeltafunc-tion.[ˆσi,ˆσj]=2iXkijkˆσk(1.7)Here[···]isthecommutationoperatorandhereistheLevi-Civitasymbol.IntheDiracproblem,wewritetheelectronspinoperatorasˆ~S,differentfromˆ~s.Andweintroduceˆ~Σ,asbelow:ˆ~S=~2ˆ~Σ(1.8)Thereareseveralpropertiesofˆ~Σ:ˆΣ2i=1(1.9){ˆΣi,ˆΣj}=0fori6=j(1.10)[ˆΣi,ˆΣj]=2iXkijkˆΣk(1.11)From(1.8)and(1.9)weobviouslygettheseproperties:ˆ~Σ·ˆ~Σ=3(1.12)ˆ~S2=34~2(1.13)AlloverthisworkweshallusetheDirac-Paulirepresentationandhencetheexplicitformsofˆ~αandˆβare:ˆ~α=0ˆ~σˆ~σ0#(1.14)ˆβ=ˆI00−ˆI(1.15)whereˆIisthe2×2unitmatrix:ˆI=10012InDirac-Paulirepresentation,ˆ~Σiswrittenexplicitlyas:ˆ~Σ=ˆ~σ00ˆ~σ#(1.16)1.3TotalangularmomentumWealreadyknowthat,asfarasDirachydrogenatomisconcerned,ˆ~J2andˆJzcommutewiththeHamiltonian.Thetotalangularmomentumoperatoris:ˆ~J=ˆ~L+ˆ~S=ˆ~L+~2ˆ~Σ=ˆ~L+~2ˆ~σ00ˆ~σ#(1.17)ˆ~J2=ˆ~L+~2ˆ~Σ2=ˆ~L2+~24ˆ~Σ2+~2(ˆ~L·ˆ~Σ+ˆ~Σ·ˆ~L)Sincetheorbitalangularmomentumandthespinareoperatingontotallydifferentspaces,theymustcommute.Also,applying(1.12)weget:ˆ~J2=ˆ~L2+34~2+~ˆ~Σ·ˆ~L=ˆ~L2+34~2+~ˆ~σ·ˆ~L00ˆ~σ·ˆ~L#(1.18)1.4Spin-angularfunctionsForspin1/2particles,wealreadyknowthatthespin-angularfunctionscanbewrittenas(Sakurai3.7.64):φAjmj=1√2l+1√l+m+1Yl,m√l−mYl,m+1(1.19)φBjmj=1√2l+3−√l−m+1Yl+1,m√l+m+2Yl+1,m+1(1.20)Theyarebothsimultaneouseigenstatesof(ˆ~J2,ˆJz),with:ˆ~J2=j(j+1)~2j=l+12(1.21)3Notethatwecannothavej=l−12becauseweusedl+1forφB,ratherthanl−1.Ifthe“l”forφBisdenotedasl0,thenl0=l+1.ˆJz=mj~mj=m+12(1.22)Finally,weknowthatφAandφBareoppositeinparity.1.5CommutationrulesWeshalldolotsofcommutationcalculationsinthiswork.Letusobtainsomeresultsforfutureuse.1.5.1Firstlookatthecommutationrulesbetweenˆ~α,ˆβandotheroperators.Weknowthatˆ~α,ˆβemergeinDiracequation,whosevaliditydoesnotdependonspace-timecoor-dinates.Becauseofthehomogeneityofspace-time,althoughˆ~αandˆβarematrices,theydonotdependonspace-timecoordinates.Hence,[ˆ~α,~r]=[ˆ~α,ˆ~p]=[ˆ~α,ˆ~L]=0(1.23)[ˆβ,~r]=[ˆβ,ˆ~p]=[ˆβ,ˆ~L]=0(1.24)Inanotherword,itisbecauseˆ~α,ˆβareoperatorsininternalspinspacethattheycommutewithanyoperatorthatonlydependsonspace-timecoordinates.1.5.2Secondly,letuswritedownthecommutationrulesbetweenˆ~Σandˆ~α,ˆβ.TheseresultsareinfactapriorrequirementforDiracequationandnoneedtobeproved.However,theexplicitmatrixforminDirac-Paulirepresentationallowsustocalculatethesecommutationrulesbymathematicalprocedure.Straightforwardasitis,thisisofnomeaninginphysicsbecausewefirstrequireitisso,andwhenwetrytoproveitagain,itmustbeso.[ˆ~Σi,ˆαi]=0(1.25)[ˆ~Σi,ˆαj]=2iXkijkˆαk(1.26)[ˆ~Σ,ˆβ]=0(1.27)41.5.3Thirdly,inDirachydrogenatomproblemwemeetwithanimportantcommutationrelation:[ˆ~L,1/r]=0.Althoughthisissomethingordinaryinquantummechanics,Iwanttoproveitherebecausethisisanimportantcharacteristicforourproblem.Westartwithtwobasicequationsincommutationcalculation.[A,BC]=B[A,C]+[A,B]C[AB,C]=A[B,C]+[A,C]BNow:0=[ˆ~p,1]=[ˆ~p,r1r]=r[ˆ~p,1r]+[ˆ~p,r]1r=r[ˆ~p,1r]−i~ˆer1rwhereˆer≡~r/r.⇒r[ˆ~p,1r]=i~ˆer1r⇒[ˆ~p,1r]=1ri~ˆer1r=i~~rr3Hence,[ˆ~L,1r]=[~r׈~p,1r]=~r×[ˆ~p,1r]+[~r,1r]׈~pObviously,[~r,1r]=0.Furth