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SJP QM 3220 3D 1
Page H-1 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
AngularMomentum(warm‐upforH‐atom)Classically,angularmomentumdefinedas(fora1‐particlesystem)
Note: definedw.r.t.anoriginofcoords.
(InQM,theoperatorcorrespondingtoLxis
accordingtoprescriptionofPostulate2,part3.)
Classically,torquedefinedas and (rotationalversionof )
Iftheforceisradial(centralforce),then H‐atom:Inamulti‐particlesystem,totalaveragemomentum:
isconservedforsystemisolatedfromexternaltorques.
sumoverparticlesInternaltorquescancauseexchangeofaveragemomentumamongparticles,but
remainsconstant.Inclassicalandquantummechanics,only4thingsareconserved:
energy linearmomentum angularmomentum electriccharge
Ox
ym
protonatorigin
electron
(Coulombforce)
SJP QM 3220 3D 1
Page H-2 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
BacktoQM.Definevectoroperator operatorunitvector
Recall
Claim:foracentralforcesuchasinH‐atom
(willshowthislater)
Thisimplies (justlikeinclassicalmechanics)
AngularmomentumofelectronisH‐atomisconstant,solongasitdoesnotabsorboremitphoton.Throughoutpresentdiscussion,weignoreinteractionofH‐atomw/photons.WillshowthatforH‐atomorforanyatom,molecule,solid–anycollectionofatoms–theangularmomentumisquantizedinunitsofħ. canonlychangebyintegernumberofħ's.
Claim:
and (i,j,kcyclic: xyzor yzxor zxy)
SJP QM 3220 3D 1
Page H-3 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
Toprove,needtwoveryusefulidentities:
Proof:
(Haveused
I'mdroppingtheˆoveroperatorswhennodangerofconfusion.Since[Lx,Ly]≠0,cannothavesimultaneouseigenstatesof
However, doescommutewithLz.
Claim:
,i=x,y,orz
Proof:
=0(Notecancellations)[L2,Lz]=0=>canhavesimultaneouseigenstatesof
allothertermslike[y,px]=0
SJP QM 3220 3D 1
Page H-4 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
LookingforwardtoH‐atom:
Wewillshowthat
=>simultaneouseigenstatesof
WhenwesolvetheTISE�ψ=EψfortheH‐atom,thenaturalcoordinatestousewillbesphericalcoordinates:r,θ,φ(notx,y,z) x=rsinθcosφ y=rsinθsinφ z=rcosθ
Justrewriting insphericalcoordinatesisgawd‐awful.But
separationofvariableswillgivespecialsolutions,energyeigenstates,offormTheangularpartofthesolutionY(θ,φ)willturnouttobeeigenstatesofL2,LzandwillhaveformcompletelyindependentofthepotentialV(r). *
energyq‐nbr
Lzq‐nbr
L2q‐nbr
y
x
z
φ
θ
SJP QM 3220 3D 1
Page H-5 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
Givenonly[L2,Lz]=0and hermiteanweknowtheremustexistsimultaneouseigenstatesf(whichwillturnouttobetheY(θ,φ)mentionedabove)suchthat (λwillberelatedtol,andμwillberelatedtom)Wewillshowthatfwilldependonquantum‐numbersl,m,sowewriteitasflm,andthat willbedeterminedlater.NoticemaxeigenvalueofLz(=lħ)issmallerthansquarerootofeigenvalueof
So,inQM,Lz<|L|Odd!Alsonoticel=0,m=0statehaszeroangularmomentum(L2=0,Lz=0)so,unlikeBohrmodel,canhaveelectroninstatethatis"justsittingthere"ratherthanrevolvingaboutprotoninH‐atom.Proofofboxedformulae:(Thisprooftakes2½pages!) DefineL+=Lx+iLy="raisingoperator"L‐=Lx‐iLy="loweringoperator"(NoteL+†=L‐,L‐†=L+,A†=hermiteanadjointofA)NeitherL+orL‐arehermitean(self‐adjoint).
Note
=>Considerf:
SJP QM 3220 3D 1
Page H-6 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
Claim:g=L+fisaneigenfunctionofLzwitheigenvalue=(μ+ħ).SoL+operatorraiseseigenvalueofLzby1ħ.Proof: ToproveLzg=(μ+ħ)g,needtoshowthat[Lz,L+]=ħL+
Now
So,operatingonfwithraisingoperatorL+raiseseigenvaluesofLZby1ħbutkeepseigenvalueofL2unchanged.(Similarly,L‐lowerseigenvalueofLzby1ħ.)OperatingrepeatedlywithL+raiseseigenvalueofLzbyħeachtime:L+(L+f)has(μ+2ħ)etc.ButeigenvalueofLzcannotincreasewithoutlimitsince cannotexceed
Thereisonlyonewayout.Theremustbeforagivenλa"topstate"ftforwhichL+ft=0.Likewise,theremustbeforagivenλa"bottomstate"fbforwhichL‐fb=0.
Lz
fb
ftL‐
L‐L+
L+
ħ
allwithsameλ=eigenvalueofL2
SJP QM 3220 3D 1
Page H-7 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
WriteLzf=mħ∙f,mchangesbyintegersonlyLzft=ℓħ∙ft,ℓ=maxvalueofmL2ft=?WanttowriteL2intermsofL+,Lz:
=> (Also, )=>
So, whereℓ=maxm,sameλforallm's.
Repeatfor minvalueofm.
(tryit!)
Sommin=‐mmaxandmchangesonlyinunitsof1.=>m=‐ℓ,‐ℓ+1,...ℓ‐2,ℓ‐1,ℓ Nintegersteps=>2ℓ=N,ℓ=N/2=>ℓ=0,1/2,1,3/2,2,5/2,...Endofproofof
SJP QM 3220 3D 1
Page H-8 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
We'llseelaterthatthereare2flavorsofangularmomentum:
m
0ℓ
1 2 3
2
‐2
‐1
0
1
‐1
0
1
0
1.OrbitalAng.Mom.(integerℓonly)
2.SpinAng.Mom.(integeror½integerOK)
SJP QM 3220 3D 1
Page H-9 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
mp>>me=>proton(nearly) stationary
Hamiltonianofelectron
TISE: specialsolutions(stationarystates).
GeneralSolutiontoTDSE:
SphericalCoordinateSystem: z=rcosθ x=rsinθcosφ y=rsinθsinφ ψ=ψ(r,θ,φ)Normalization:Need insphericalcoordinates
HardWay:
TheHatom
mp≈1840me
me
z
φ
θ
r
volume
SJP QM 3220 3D 1
Page H-10 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
Alsoneed9derivatives:
EasierWay:Curvilinearcoordinates(SeeBoas)pathelement:
Sphericalcoordinates:
*InClassicalMechanics(CM),KE=p2/2m=KE=(radialmotionKE)+(angular,axialmotionKE)O
SJP QM 3220 3D 1
Page H-11 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
*SamesplittinginQM:
(Notice dependsonlyonθ,φandnotr.)
SeparationofVariables!(asusual)Seekspecialsolutionofform:
Normalization:∫dV|ψ|2=(Convention:normalizeradial,angularpartsindividually)Plugψ=R∙YintoTISE=>
Multiplythruby :
=>f(r)=g(θ,φ)=constantC=ℓ(ℓ+1)
SJP QM 3220 3D 1
Page H-12 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
HaveseparatedTISEintoradialpartf(r)=ℓ(ℓ+1),involvingV(r),andangularpartg(θ,φ)=ℓ(ℓ+1)whichisindependentofV(r).=>Allproblemswithsphericallysymmetricpotential(V=V(r))haveexactlysameangularpartofsolution:Y=Y(θ,φ)called"sphericalharmonics".We'lllookatangularpartlater.Now,let'sexamine
RadialSE:
Changeofvariable:u(r)=r∙R(r)
Canshowthat
Notice:identicalto1DTISE:
except
r:0‐>∞insteadofx:‐∞‐>+∞and
V(x)replacedwith
Veff="effectivepotential"Boundaryconditions:
same!
SJP QM 3220 3D 1
Page H-13 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
Seek bound state solutions E < 0
E > 0 solutions are unbound states, scattering solutions
u(r=∞)=0fromnormalization∫dr|u|2=1
u(r=0)=0,otherwise blowsupatr=0(subtle!)
FullsolutionofradialSEisverymessy,eventhoughitiseffectivelya1Dproblem(differentproblemforeachℓ)Powerseriessolution(seetextfordetails).Solutionsdependon2quantumnumbers:nandℓ(foreacheffectivepotentialℓ=0,1,2,…haveasetofsolutionslabeledbyindexn.)Solutions:n=1,2,3,… forgivenn ℓ=0,1,…(n‐1) ℓmax=(n–1)n="principalquantumnumber"energyeigenvaluesdependonnonly(itturnsout)
(independentofℓ)
•sameasBohrmodel,agreeswithexperiment!
Notice that energy eigenvalues given by solution to radial equation alone.
SJP QM 3220 3D 1
Page H-14 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
Firstfewsolutions:Rnℓ(r)normalization"Bohrradius"
NOTE:•forℓ=0(sstates),R(r=0)≠0=>wavefunctionψ"touches"nucleus.•forℓ≠0,R(r=0)=0=>ψdoesnottouchnucleus.ℓ≠0=>electronhasangularmomentum.Sameasclassicalbehavior,particlewithnon‐zeroLcannotpassthruorigin CanalsoseethisinQM:forℓ≠0,Veffhasinfinitebarrieratorigin=>u(r)mustdecaytozeroatr=0exponentially.
=>exponentialdecayin
aswell.
SJP QM 3220 3D 1
Page H-15 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
Backtoangularequation: Wanttosolveforthe ‐"sphericalharmonics".Before,startedwithcommutationrelations,
and,usingoperatoralgebra,solvedfortheeigenvaluesofL2,Lz.Wefound whereℓ=0,½,1,3/2,… m=‐ℓ,‐ℓ+1…+ℓIntheprocess,wedefinedraisingandloweringoperators:
(cmissomeconstant)
So,ifwecanfind(foragivenℓ)asingleeigenstate ,thenwecangenerateallthe
others(otherm's)byrepeatedapplicationof .
It'seasytofindtheφ‐dependence;don'tneedthe businessyet.
€
ˆ L z =
i∂∂ϕ
(showed in HW)
ˆ L zY =
i∂Y∂ϕ
= mY (and you can cancel the )
Assume
Ifweassume(postulate)thatψissingle‐valuedthan=>m=0,±1,±2,…Butm=‐ℓ,…+ℓSofororbitalangularmomentum,ℓmustbeintegeronly:ℓ=0,1,2,…(throwout½integervalues)
SJP QM 3220 3D 1
Page H-16 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
* (algebra!)
Candeduce from
=>
Solution:(un‐normalized)Checks:Plugbackin.
Now,cangetother byrepeatedapplicationof Somewhatmessy(HW!)
Normalizationfrom Noticecaseℓ=0 :Example:Conventionon±sign:
SJP QM 3220 3D 1
Page H-17 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
Thesphericalharmonicsformacomplete,orthonormalset(sinceeigenfunctionsofhermiteanoperators)Anyfunctionofanglesf=f(θ,φ)canbewrittenaslinearcomboof :Likewise:
=>H‐atomenergyeigenstatesare n=1,2,…;ℓ=0,1…(n‐1);m=‐ℓ…+ℓArbitrary(bound)stateis (c'sareanycomplexconstants)energyofstate(n,ℓ,m)dependsonlyonn.En=‐constant/n2(statesℓ,mwithsamenaredegenerate)
Degeneracyofnthlevelisn2(2•n2ifyouincludespin)
ℓ=
n=
2
3d
1
4d(1)4s
4f
3
4
0 1 2 3
3s2s1s
4p
3p
2p
(3) (5) (7)
SJP QM 3220 3D 1
Page H-18 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
Prob(findparticleindVabout )=
Ifℓ=0,ψ=ψ(r)then Prob(findinr�r+dr)=P(r)=radialprobabilitydensity
Groundstate:
NoticeP(r)verydifferentfromψ(r):
Ifℓ≠0,ψ=ψ(r,θ,φ)=R(r)Y(θ,φ),then
Prob(findinr�r+dr)=r2|R|2dr evenifℓ≠0Note: ifH‐atomandemission/absorptionofradiation:IfH‐atomisinexcitedstate(n=2,ℓ=1,m=0)thenitisinenergyeigenstate=stationarystate.Ifatomisisolated,thenatomshouldremaininstateψ210forever,sincestationarystatehassimpletimedependence:
RadialProbabilityDensity
"solidangle"
P(r)=r2|R|2
SJP QM 3220 3D 1
Page H-19 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
But,experimentally,wefindthatH‐atomemitsphotonandde‐excites:ψ210‐>ψ100in≈10‐7s‐>10‐9s
Thereasonthattheatomdoesnotremaininstationarystateisthatitisnottrulyisolated.TheatomfeelsafluctuatingEMfielddueto"vacuumfluctuations".QuantumElectrodynamicsisarelativistictheoryoftheQMinteractionofmatterandlight.Itpredictsthatthe"vacuum"isnot"empty"or"nothing"aspreviouslysupposed,butisinsteadaseethingfoamofvirtualphotonsandotherparticles.ThesevacuumfluctuationsinteractwiththeelectronintheH‐atomandslightlyalterthepotentialV(r).Soeigenstatesofthecoulombpotentialarenoteigenstatesoftheactualpotential:Vcoulomb+VvacuumPhotonspossessanintrinsicangularmomentum(spin)of1ħ,meaningSowhenanatomabsorbsoremitsasinglephoton,itsangularmomentummustchangeby1ħ,byConservationofAngularMomentum,sotheorbitalangularmomentumquantumnumberℓmustchangeby1."SelectionRule":∆ℓ=±1inanyprocessinvolvingemissionorabsorptionof1photon=>allowedtransitionsare:
IfanH‐atomisinstate2s(n=2,ℓ=0)thenitcannotde‐excitetogroundstatebyemissionofaphoton.(sincethiswouldviolatetheselectionrule).Itcanonlyloseitsenergy(de‐excite)bycollisionwithanotheratomorviaarare2‐photonprocess.
ps
n=1
2
3
4
5d
2p
1s
E
Eγ=hf=∆E
SJP QM 3220 3D 1
Page H-20 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
completeorthonormalset
€
ψ = cnn∑ n , ψ → cn{ } =
c1
c2
c3
cn
€
u1 =
100
u2 =
010
Ifket'sarerepresentedbycolumnvectors,thenbra'sarerepresentedbythetransposeconjugateofcolumn=row,complexconjugate.
Operatorscanberepresentedbymatrices: nohatonmatrixelement
where{|n>}issomecompleteorthonormalset.
MatrixFormulationofQM
SJP QM 3220 3D 1
Page H-21 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
Whyisthat?Wheredoesthatmatrixcomefrom?Considertheoperator and2statevectors relatedby
() Inbasis{|n>},
€
ψ = cnn∑ n = n
n∑ n ψ
cn
ϕ = dnn∑ n = n
n∑ n ϕ
dn
Nowprojectequationonto|m>byactingwithbra:
€
m ϕ = m ˆ A ψ = cnn∑ m ˆ A n
dn = Amnn∑ cn
But,thisissimplytheruleformultiplicationofmatrixcolumn.
€
d1
d2
d3
=
A11 A12 …
A21 A22
c1
c2
Sothereyouhaveit,that'swhytheoperatorisdefinedasthismatrix,inthisbasis!Now,suppose areenergyeigenstates,then
Amatrixoperator isdiagonalwhenrepresentedinthebasisofitsowneigenstates,andthediagonalelementsaretheeigenvalues.Noticethatingeneraloperatorsdon'tcommute .SamegoesforMatrixMultiplication:A B≠B A
SJP QM 3220 3D 1
Page H-22 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
Claim:Thematrixofahermitianoperatorisequaltoitstransposeconjugate:Proof:
€
m ˆ A n = ˆ A m n = n ˆ A m*
⇒ Amn = Anm*
Similarly,adjoint(or"Hermitianconjugate") Proof:
€
ˆ A m n = m ˆ A tn = n ˆ A m*
Ofcourse,it'sdifficulttodocalculationsifthematricesandcolumnsareinfinitedimensional.ButthereareHilbertsubspacesthatarefinitedimensional.Forinstance,intheH‐atom,thefullspaceofboundstatesisspannedbythefullset{n,ℓ,m}(=|nℓm>).Thesub‐set{n=2,ℓ=1,m=+1,0,‐1}formsavectorspacecalledasubspace.Subspace?InordinaryEuclideanspace,anyplaneisasubspaceofthefullvolume.Ifweconsiderjustthexycomponentsofavector ,thenwehaveaperfectlyvalid2Dvectorspace,eventhoughthe"true"vectoris3D.Likewise,inHilbertspace,wecanrestrictourattentiontoasubspacespannedbyasmallnumberofbasisstates.Example:H‐atomsubspace{n=2,ℓ=1,m=+1,0,‐1}Basisstatesare (candropn=2,ℓ=1inlabelsincetheyarefixed.)
€
ˆ L z m = m m
ˆ L 2 m = 2( +1) m
( =1)= 22 m (for all m)
€
⇒ Lz( )mn= m ˆ L z n =
+1 0 00 0 00 0 −1
SJP QM 3220 3D 1
Page H-23 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
€
L2mn = m ˆ L 2 n = 2
1 0 00 1 00 0 1
(WhataboutLx?Ly?)Beforeseeingwhatallthismatrixstuffisgoodfor,let'sexaminespinbecauseit'sveryimportantphysicallyandbecauseitwillleadto2DHilbertspacewithsimple2x2matrices.bracketorinner‐product:
Whichintegralyoudodependsontheconfigurationspaceofproblem.Keydefiningpropertiesofbracket:
<f|g>*=<g|f> c=constant
<f|c•g>=c<f|g>,<c•f|g>=c*<f|g>
<α|(b|β>+c|γ>)=b<α|β>+c<α|γ>Diracproclaims:<g|f>=<g|nextto|f> bracket="bra"and"ket"Ket|f>representsvectorinH‐space(HilbertSpace)"ket""wavefunction"
Bothψandψ(x)describesamestate,but|ψ>ismoregeneral:
ReviewofDiracBraKetNotation
Different"representations"ofsameH=spacevector|ψ>
SJP QM 3220 3D 1
Page H-24 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
�position‐representation,momentum‐rep,energy‐rep.
SJP QM 3220 3D 1
Page H-25 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
Whatisa"bra"?<g|isanewkindofmathematicalobject,calleda"functional".
insertstatefunctionhere input outputfunction: number numberoperator: function functionfunctional: function numbe<g|wantstobindwith|f>toproduceinnerproduct<g|f>Foreveryket|f>thereisacorrespondingbra<f|.Likethekets,thebra'sformavectorspace.
|cf>�<cf|=c*<f| (?)
|αf+βg>�<αf+βg|=α*<f|+β*<g|
<αf+βg|h>=α*<f|h>+β*<g|h>
Complexnumberbra=anotherbra =>bra'sformanylinearcomboofbra's=anotherbra vectorspaceThevectorspaceofbrasiscalleda"dualspace".It'sthedualoftheketvectorspace.
isaket.Whatisthecorrespondingbra?
Definition:hermiteanconjugateoradjoint
€
ˆ A f g ≡ f Atg forallf,g.
(If ishermiteanorself‐adjoint.)Someproperties:
€
Proof : f ˆ A t( )tg = ˆ A t f g =
g ˆ A t f*
= ˆ A g f*
= f ˆ A f
Def'nofA†
SJP QM 3220 3D 1
Page H-26 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
Theadjointofanoperatorisanalogoustocomplexconjugateofacomplexnumber:
The"ket‐bra"|f><g|isanoperator.Itturnsaket(function)intoanotherket(function):
€
f g ( ) h = f g h ProjectionOperators
€
ψ(x) = cnun (x) = un ψ n∑
n∑ un (x)→
ψ = cn nn∑ = n ψ
n∑ n = n
n∑ n ψ
=> "Completenessrelation" (discretespectrumcase)
="projectionoperator"
picksoutportionofvector|ψ>thatliesalong|n>
€
ˆ P n ψ = n n ψ = cn n
|ψ>
u2=|2>
u1=|1>
|2><2|ψ>
u1<u1|ψ>=|1><1|ψ>
SJP QM 3220 3D 1
Page H-27 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
€
ψ = nn∑ n ψ like
R = ˆ x x ⋅
R ( ) + ˆ y ˆ y ⋅
R ( )
€
ˆ 1 = nn∑ n like 1 = ˆ x x ⋅ _ ( ) + ˆ y ˆ y ⋅ _ ( )
Anywherethereisaverticalbarinthebracket,oraketorabra,wecanreplacethebarwith
Example:
=>
Ifeigenvaluespectrumiscontinuous(asfor )thenmustuseintegral,ratherthansum,overstates.
CompletenessRelation(continuousspectrum)
Example:
€
Φ(p) = f p ψ = dx∫ f p x x ψ =1
2πdx∫ e− ipx ψ(x)
TheMeasurementPostulates3and4canberestatedintermsoftheprojectionoperator:Startingwithstate ,
wheresum{n}isoveranycompletesetofstates,ifwemeasureobservableassociatedwithn,thenwewillfindvaluen0withprobability
Probabilityoffindingeigenvaluen0=expectationvalueofprojectionoperator .Andasresultofmeasurementstate|ψ>collapsestostate . (apartfromnormalization)
SJP QM 3220 3D 1
Page H-28 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
Wecannowgeneralizetocaseofstatesdescribedbymorethanoneeigenvalue,suchasH‐atom.
Ifwemeasureenergy(butnotalso ),findn0,thenweareprojectingonto
subspacespannedby{ℓ,m}withsomen0.
Statecollapsesto
mustrenormalize
SJP QM 3220 3D 1
Page H-29 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
Spin½RecallthatintheH‐atomsolution,weshowedthatthefactthatthewavefunction
Ψ(r)issingle‐valuedrequiresthattheangularmomentumquantum#beinteger:l=
0,1,2..However,operatoralgebraallowedsolutionsl=0,1/2,1,3/2,2…
Experimentshowsthattheelectronpossessesanintrinsicangularmomentum
calledspinwithl=½.Byconvention,weusethelettersinsteadofl forthespin
angularmomentumquantumnumber:s=½.
Theexistenceofspinisnotderivablefromnon‐relativisticQM.Itisnotaformof
orbitalangularmomentum;itcannotbederivedfrom .
(Theelectronisapointparticlewithradiusr=0.)
Electrons,protons,neutrons,andquarksallpossessspins=½.Electronsand
quarksareelementarypointparticles(asfaraswecantell)andhavenointernal
structure.However,protonsandneutronsaremadeof3quarkseach.The3half‐
spinsofthequarksaddtoproduceatotalspinof½forthecompositeparticle(ina
sense,↑↑↓makesasingle↑).Photonshavespin1,mesonshavespin0,thedelta‐
particlehasspin3/2.Thegravitonhasspin2.(Gravitonshavenotbeendetected
experimentally,sothislaststatementisatheoreticalprediction.)
SJP QM 3220 3D 1
Page H-30 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
SpinandMagneticMoment
Wecandetectandmeasurespinexperimentallybecausethespinofa
chargedparticleisalwaysassociatedwithamagneticmoment.
Classically,amagneticmomentisdefinedasavectorµ associatedwith
aloopofcurrent.Thedirectionofµ isperpendiculartotheplaneof
thecurrentloop(right‐hand‐rule),andthemagnitudeis
.
Theconnectionbetweenorbitalangularmomentum(notspin)andmagnetic
momentcanbeseeninthefollowingclassicalmodel:Consideraparticlewithmass
m,chargeqincircularorbitofradiusr,speedv,periodT.
|angularmomentum|=L=pr=mvr,sovr=L/m,and .
Soforaclassicalsystem,themagneticmomentisproportionaltotheorbital
angularmomentum: .
Thesamerelationholdsinaquantumsystem.
InamagneticfieldB,theenergyofamagneticmomentisgivenby
(assuming ).InQM, .
Writingelectronmassasme(toavoidconfusionwiththemagneticquantumnumber
m)andq=–ewehave ,wherem=−l..+l.Thequantity
ri
µ
r
i
m,q
SJP QM 3220 3D 1
Page H-31 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
iscalledtheBohrmagneton.Thepossibleenergiesofthemagnetic
momentin isgivenby .
Forspinangularmomentum,itisfoundexperimentallythattheassociatedmagnetic
momentistwiceasbigasfortheorbitalcase:
(WeuseSinsteadofLwhenreferringtospinangularmomentum.)
Thiscanbewritten .
Theenergyofaspininafieldis (m=±1/2)afactwhichhasbeen
verifiedexperimentally.
Theexistenceofspin(s=½)andthestrangefactorof2inthegyromagneticratio
(ratioof )wasfirstdeducedfromspectrographicevidencebyGoudsmitand
Uhlenbeckin1925.
Another,evenmoredirectwaytoexperimentallydeterminespiniswithaStern‐
Gerlachdevice,nextpage
SJP QM 3220 3D 1
Page H-32 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
(ThispagefromQMnotesofProf.RogerTobin,PhysicsDept,TuftsU.)
Stern-Gerlach Experiment (W. Gerlach & O. Stern, Z. Physik 9, 349-252 (1922).
€
F = ∇ µ • B ( ) = ( µ •
∇ ) B (in current free regions), or here,
F = ˆ z (µz
∂Bz
∂z)(thisisalittle
crude‐seeGriffithsExample4.4forabettertreatment,butthisgivesthemainidea)
Deflectionofatomsinz‐directionisproportionaltoz‐componentofmagnetic
momentµz,whichinturnisproportionaltoLz.Thefactthattherearetwobeamsis
proofthatl=s=½.Thetwobeamscorrespondtom=+1/2andm=–1/2.Ifl=1,
thentherewouldbethreebeams,correspondingtom=–1,0,1.Theseparationof
thebeamsisadirectmeasureofµz,whichprovidesproofthat
Theextrafactorof2intheexpressionforthemagneticmomentoftheelectronis
oftencalledthe"g‐factor"andthemagneticmomentisoftenwrittenas
.Asmentionedbefore,thiscannotbededucedfromnon‐relativistic
QM;itisknownfromexperimentandisinserted"byhand"intothetheory.
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Page H-33 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
However,arelativisticversionofQMduetoDirac(1928,the"DiracEquation")
predictstheexistenceofspin(s=½)andfurthermorethetheorypredictsthevalue
g=2.Alater,betterversionofrelativisticQM,calledQuantumElectrodynamics
(QED)predictsthatgisalittlelargerthan2.Theg‐factorhasbeencarefully
measuredwithfantasticprecisionandthelatestexperimentsgiveg=
2.0023193043718(±76inthelasttwoplaces).ComputingginQEDrequires
computationofabinfiniteseriesoftermsthatinvolveprogressivelymoremessy
integrals,thatcanonlybesolvedwithapproximatenumericalmethods.The
computedvalueofgisnotknownquiteaspreciselyasexperiment,neverthelessthe
agreementisgoodtoabout12places.QEDisoneofourmostwell‐verified
theories.
SpinMath
Recallthattheangularmomentumcommutationrelations
werederivedfromthedefinitionoftheorbitalangularmomentumoperator:
.
Thespinoperator doesnotexistinEuclideanspace(itdoesn'thaveapositionor
momentumvectorassociatedwithit),sowecannotderiveitscommutation
relationsinasimilarway.Insteadweboldlypostulatethatthesamecommutation
relationsholdforspinangularmomentum:
.Fromthese,wederive,justabefore,that
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Page H-34 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
(sinces=½)
(sincems=−s,+s=−1/2,+1/2)
Notation:sinces=½always,wecandropthisquantumnumber,andspecifythe
eigenstatesofL2,Lzbygivingonlythemsquantumnumber.Therearevariousways
towritethis:
Thesestatesexistina2DsubsetofthefullHilbertSpacecalledspinspace.Since
thesetwostatesareeigenstatesofahermitianoperator,theyformacomplete
orthonormalset(withintheirpartofHilbertspace)andany,arbitrarystateinspin
spacecanalwaysbewrittenas (Griffiths'notationis
)
Matrixnotation: .Notethat
IfwewereworkinginthefullHilbertSpaceof,say,theH‐atomproblem,thenour
basisstateswouldbe .Spinisanotherdegreeoffreedom,sothatthe
fullspecificationofabasisstaterequires4quantumnumbers.(Moreonthe
connectionbetweenspinandspacepartsofthestatelater.)
[Noteonlanguage:throughoutthissectionIwillusethesymbolSz(andSx,etc)to
refertoboththeobservable("themeasuredvalueofSzis ")anditsassociated
operator("theeigenvalueofSzis ").]
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Page H-35 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
ThematrixformofS2andSzinthe basiscanbeworkedoutelementby
element.(Recallthatforanyoperator .)
Operatorequationscanbewritteninmatrixform,forinstance,
WearegoingaskwhathappenswhenwemakemeasurementsofSz,aswellasSx
andSy,(usingaStern‐Gerlachapparatus).Willneedtoknow:Whatarethe
matricesfortheoperatorsSxandSy?Thesearederivedfromtheraisingand
loweringoperators:
TogetthematrixformsofS+,S−,weneedaresultfromthehomework:
Forthecases=½,thesquarerootfactorsarealways1or0.Forinstance,s=½,
m=−1/2gives .Consequently,
,leadingto
and
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Page H-36 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
NoticethatS+,S−arenothermitian.
Using yields
Thesearehermitian,ofcourse.
Oftenwritten: ,where are
calledthePaulispinmatrices.
Nowlet'smakesomemeasurementsonthestate .
Normalization: .
SupposewemeasureSzonasysteminsomestate .
Postulate2saysthatthepossibleresultsofthismeasurementareoneoftheSz
eigenvalues: .Postulate3saystheprobabilityoffinding,say ,
is .
Postulate4saysthat,asaresultofthismeasurement,whichfound ,theinitial
state collapsesto .
ButsupposewemeasureSx?(WhichwecandobyrotatingtheSGapparatus.)
Whatwillwefind?Answer:oneoftheeigenvaluesofSx,whichweshowbeloware
thesameastheeigenvaluesofSz: .(Notsurprising,sincethereis
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Page H-37 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
nothingspecialaboutthez‐axis.)Whatistheprobabilitythatwefind,say,Sx=
?ToanswerthisweneedtoknowtheeigenstatesoftheSxoperator.Let's
callthese(sofarunknown)eigenstates (Griffithscallsthem
).Howdowefindthese?Wemustsolvetheeigenvalueequation:
,whereλ aretheunknowneigenvalues.Inmatrixformthisis,
whichcanberewritten .In
linearalgebra,thislastequationiscalledthecharacteristicequation.
Thissystemoflinearequationsonlyhasasolutionif
.So
Asexpected,theeigenvaluesofSxarethesameasthoseofSz(orSy).
Nowwecanplugineacheigenvalueandsolvefortheeigenstates:
; .
Sowehave
Nowbacktoourquestion:Supposethesysteminthestate ,andwe
measureSx.Whatistheprobabilitythatwefind,say,Sx= ?Postulate3gives
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Page H-38 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
therecipefortheanswer:
Questionforthestudent:Supposetheinitialstateisanarbitrarystate
andwemeasureSx.WhataretheprobabilitiesthatwefindSx= and ?
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Page H-39 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
Let'sreviewthestrangenessofQuantumMechanics.
SupposeanelectronisintheSx= eigenstate .Ifweask:What
isthevalueofSx?Thenthereisadefiniteanswer: .Butifweask:Whatisthe
valueofSz,thenthisisnoanswer.ThesystemdoesnotpossessavalueofSz.Ifwe
measureSz,thentheactofmeasurementwillproduceadefiniteresultandwillforce
thestateofthesystemtocollapseintoaneigenstateofSz,butthatveryactof
measurementwilldestroythedefinitenessofthevalueofSx.Thesystemcanbein
aneigenstateofeitherSxorSz,butnotboth.