TFY-44.130 Kvanttimekaniikka II electromagnetic (EM) field can be ...

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18 Οκτ 2013 (πριν από 3 χρόνια και 10 μήνες)

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TFY-44.130KvanttimekaniikkaII
S.Virtanen2007
electromagnetic(EM)eldcanbewritteninthepositionrepresentationas
H=
N
X
i=1

1
2m
h
ˆp
i

q
c
A(r
i
,t)
i
2
+qφ(r
i
,t)−
Zq
2
r
i

ˆ
M
si
B(
ˆr,t)

+
1
2
N
X
i=1
X
j6=i
q
2
|r
i
−r
j
|
,
(3.19)
wheremistheelectronmass,q=−eitscharge,r
i
thepositionoperatorof
electroni,
ˆ
p
i
=
￿
i

i
itsmomentumoperatorand
ˆ
M
si
itsspinmagnetic
momentoperator,andZthechargenumberofthenucleus.Theclassical
eldsA(r,t),φ(r,t)arethevectorandscalarpotentialsforthe
electromagneticeld:
5
E=−∇φ−
1
c
∂A
∂t
,(3.20)
B=∇×A.(3.21)
Theterms−
ˆ
M
si
B(
ˆr,t)representtheinteractionoftheelectronspinwith
themagneticeld:
ˆ
M
s
=−g
s

B
￿
ˆS=g
s
q
2mc
ˆ
S≈
q
mc
ˆS(3.22)
isthespinmagneticmomentoperatorofanelectron,g
s
≈2.002the
gyromagneticratiooftheelectron,
B
=
e￿
2mc
theBohrmagnetonand
ˆ
Sthe
spinoperatoroftheelectron.
Itiseasytobelievethatsolvingtheexacttime-developmentoftheatom
usingthisHamiltonianisratherdifcult.
Thuswemakethefollowingsimplications/approximations:
•Weassumethattheradiationisnottoostrongsuchthatrst-order
perturbationtheorycanbeused.Thisissensiblebecause
electromagneticeldsofalasersaretypicallymuchweakerthanthe
eldsinsideofatoms.
5
WeusetheGaussiansystemofunits.
54
TFY-44.130KvanttimekaniikkaII
S.Virtanen2007
•Weneglectthecorrelationsandinteractionsbetweenelectrons,anduse
thefactthatfornon-Xradiationitisthevalenceelectron(s)thatmake
transitions,andtheeffectontheinnerelectronscanusuallybe
neglected.Theproblemthusreducestoanalysingthephysicsoflet's
sayoneelectronmovingundertheCoulombattractionofanucleusthat
haseffectivechargeZ

e(recallthevariationalcalculationoftheground
stateenergyofheliuminKvanttimekaniikkaI).Thismodelshouldbe
sensibleatleastforalkaliatomswithonevalenceelectron.
Thesesimplications/approximationsamounttomodelingthephysicswith
theeffectiveone-electronHamiltonian
H=
1
2m
h
ˆ
p−
q
c
A(
ˆr,t)
i
2
+qφ(
ˆr,t)−
Z

e
2
r

ˆ
M
s
B(
ˆr,t).(3.23)
Touseperturbationtheory,wewritethisHamiltonianas
H=H
0
+H

(t),(3.24)
where
H
0
=
ˆ
p
2
2m

Z

e
2
r
(3.25)
isthetime-independentHamiltonianofthehydrogen-likefreeatom,andthe
time-dependentperturbation
H

=−
q
2mc
(
ˆ
pA+A
ˆ
p)+
q
2
2mc
2
A
2
+qφ−
ˆ
M
s
B(3.26)
describestheinteractionoftheatomwiththeradiationeld.Notethatthe
eigenstatesofH
0
arethethefamiliarone-electron|φ
nlm
istates,butpossibly
correspondingtoZ

6=1(forZ

=1theyarejusttheeigenstatesof
hydrogen).
TheelectromagneticeldsandtheSchrödingerequation
i￿
∂ψ
∂t
=Hψ(3.27)
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TFY-44.130KvanttimekaniikkaII
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areinvariantundergaugetransformationsoftheform
A

=A+∇χ,(3.28)
φ

=φ−
1
c
∂χ
∂t
,(3.29)
ψ

=ψe
i(q/￿c)χ
,(3.30)
whereχ=χ(r,t)isanarbitraryscalarfunction(Verifythis!).Thegauge
degreeoffreedommaybexedatwill,andcalculationscanbeoften
simpliedconsiderablybyajudiciouschoiceofgauge.Inthefollowingwe
usetheCoulombgauge,inwhichthegaugeisxedbythegaugecondition
∇A=0.(3.31)
TheMaxwellequationsandthisgaugeconditionimply

2
φ=4πρ,(3.32)
whereρ=ρ(r,t)isthechargedistribution.Sinceweneglecttheeffectof
theatomtotheEMeld,ρ(r,t)≡0andwemaychoose
φ(r,t)≡0.(3.33)
Letusrsttaketheperturbingexternalelectromagneticeldtobea
monocromaticplaneelectromagneticwaveinsection3.3.2wediscussthe
caseofnonmonochromaticradiation.IntheCoulombgauge,thisis
representedbythevectorpotential
A(r,t)=[A
0
e
i(kr−ωt)
+A

0
e
−i(kr−ωt)
]ǫ,(3.34)
whereǫisaunitpolarizationvectorandkthewavevectorpointingtothe
propagationdirection.ThedispersionrelationforEMwavesisω=ck.The
transversalitycondition
ǫk=0(3.35)
alwayssatisedbyfreelypropagatingEMwavesalsoimpliesthegauge
condition(3.31)tobesatised.
Ontheotherhand,dueto(3.31)and(3.33),ourtime-dependentperturbation
H

(t)reducestotheform
6
H

=−
q
mc
A
ˆ
p+
q
2
2mc
2
A
2

ˆ
M
s
B(3.36)
6
Recallthat
ˆpA=
￿
i
∇A,and∇Aψ=[∇A]ψ+A∇ψ.
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TFY-44.130KvanttimekaniikkaII
S.Virtanen2007
intheCoulombgauge.
Infact,thetwolasttermsmaytypicallybeneglectedtogoodapproximation:
•Theintensityoftheradiationisproportionalto|A
0
|
2
,andtheintensity
ofusuallightsourcesislowenoughsuchthat
q
2
2mc
2
A
2

q
2
2mc
2
|A
0
|
2
is
negligiblecomparedtotherstterm.
7
•Alsowemaymakethecrudeorderofmagnitudeestimate




hf|
ˆ
M
s
B|ii
hf|
q
mc
(A
ˆ
p)|ii









q
mc
SB
q
mc
A
0
p









￿kA
0
A
0
p




=
￿k
p
,(3.37)
sinceS∼￿andB∼kA
0
duetoB=∇×A.Nowthexp-uncertainty
relationsuggeststhat￿/p￿a
0
,wherea
0
istheBohrradius
characterizingtheradiusoftheatom.Wethusnd




hf|
ˆ
M
s
B|ii
hf|
q
mc
(A
ˆ
p)|ii





a
0
λ
,(3.38)
whereλ=2π/kisthewavelengthoftheradiation.Intheopticalregion
λ∼500nm,andbecausea
0
≈0.5Å=0.05nm,itfollows
a
0
λ
≈0.0001.(3.39)
Thethirdtermisthusintheopticalregionalsonegligiblecomparedto
therstone!
Notethatforcertainstatesthematrixelementhf|
q
mc
(A
ˆ
p)|iimay
vanishidentically,andthenofcourseourorderofmagnitudeestimate
fails.Infactthishappensquiteoftenduetosymmetryeffects,andwill
leadtoselectionrulesfortransitions(wewillcomebacktothisabit
later).Insuchacasethedominating(althoughsmall)contributionis
givenbytheotherterms.However,whenhf|
q
mc
(A
ˆp)|iiisnonzero,
ourorderofmagnitudeestimateshowsthatittypicallygivesthe
dominatingcontribution.
7
TheradiationeldisusuallymuchweakerthantheverystronginternalEMeldswithintheatom.Thisisessentialfor
ourperturbationaltreatmentintherstplace.
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TFY-44.130KvanttimekaniikkaII
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Consequently,intheopticalregionitshouldbeagoodapproximationtotake
simply
H

≈−
q
mc
A
ˆ
p=−
q
mc
[A
0
e
i(kr−ωt)
+A

0
e
−i(kr−ωt)
]ǫ
ˆp(3.40)
astheperturbationHamiltonian.
Now,assumingthewavetohaveinuencedtheatomoverseveralfrequency
periods(i.e.ωt≫1),wecanapplythegeneralresult(3.12),ndingforthe
electronstateoftheatomthetransitionprobabilities
P
if
(t)=













q
2
|A
0
|
2
m
2
c
2
￿
2
|hf|e
−ikr
ǫ
ˆ
p|ii|
2

sin
[
1
2
(ω+ω
fi
)t
]
1
2
(ω+ω
fi
)

2
,forω
fi
≈−ω<0
q
2
|A
0
|
2
m
2
c
2
￿
2
|hf|e
ikr
ǫ
ˆ
p|ii|
2

sin
[
1
2
(ω−ω
fi
)t
]
1
2
(ω−ω
fi
)

2
,forω
fi
≈+ω>0
0,otherwise.
(3.41)
Thetime-averagedenergydensityoftheplanewaveEMeldcanbeshown
tobe
ρ
E
=
1

(E
2
+B
2
)=
ω
2
2πc
2
|A
0
|
2
,(3.42)
sothatwecanexpressourresultintheform
P
if
(t)=













2πq
2
ρ
E
m
2
￿
2
ω
2
|hf|e
−ikr
ǫ
ˆ
p|ii|
2

sin
[
1
2
(ω+ω
fi
)t
]
1
2
(ω+ω
fi
)

2
,forω
fi
≈−ω<0
2πq
2
ρ
E
m
2
￿
2
ω
2
|hf|e
ikr
ǫ
ˆp|ii|
2

sin
[
1
2
(ω−ω
fi
)t
]
1
2
(ω−ω
fi
)

2
,forω
fi
≈+ω>0
0,otherwise.
(3.43)
IntherstcaseE
f
−E
i
=￿ω
fi
<0,i.e.theenergyoftheatomisreduced
inthetransition.Accordingly,itcorrespondstoastimulatedemission
process,inwhichtheelectromagneticeldstimulatestheatomtoemit
energy.Actually,theatomemitsenergybyphotonemission,buttheemitted
photonisnotseeninourformalismwhichtreatstheEMeldclassically.
Correspondingly,inthesecondcasetheE
f
−E
i
=￿ω
fi
>0,i.e.theenergy
oftheatomisincreasedinthetransition.Theprocesscorrespondstoan
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TFY-44.130KvanttimekaniikkaII
S.Virtanen2007
absorptionofenergybytheatomfromtheEMeld.Actually,theatom
absorpsaphotoninsuchaprocess.
3.3.1Electricdipoletransitions;Selectionrules
Inordertondoutthetransitionrates,westillhavetocalculatethematrix
elements
M
fi
=hf|e
±ikr
ǫ
ˆ
p|ii(3.44)
thatappearin(3.43).Theinitialandnalstatesareenergyeigenstatesofan
unperturbedhydrogenicatom.
Wenotethatthewavefunctionsoftheatomicstatesessentiallyvanish
outsidefewBohrradiia
0
.Since
|kr|￿
a
0
λ
≈0.0001(3.45)
intheregioninwhichthewavefunctionsarenonzerowemayapproximate
e
±ikr
=1±ikr+≈1(3.46)
withgoodaccuracyinthematrixelements:
M
fi
≈hf|ǫ
ˆ
p|ii=ǫhf|
ˆ
p|ii.(3.47)
Thisistheso-calledelectricdipoleapproximation;notethatwithinitthe
matrixelementsdonotdependatallonthewavevectorkoftheradiation.
Theapproximationmaybeinterpretedasfollows:Theatomismuchsmaller
thantheopticalwavelengthoftheEMeld,andthustheexternaleldis
essentiallyconstantwithintheatomicvolume.Bytakingintoaccounthigher
ordertermsintheexpansionofe
±ikr
,onegetsmagneticdipole,electric
quadrupoleetc.matrixelements,whicharecorrectionstotheelectricdipole
approximation.Thesecorrectionsareimportantiftheelectricdipolematrix
elementvanishes.Note,however,thatinhigherordercalculationsalsothe
secondandthirdtermsintheinteractionHamiltonian(3.36)havetobetaken
intoaccounttobeconsistent.
Nowsince
[H
0
,ˆx]=[
ˆ
p
2
2m
+V(
ˆr),ˆx]=
1
2m
2[
ˆ
p,ˆx]
ˆp=
1
2m
2(−i￿e
x
)
ˆ
p=
￿
im
ˆp
x
,
(3.48)
59
TFY-44.130KvanttimekaniikkaII
S.Virtanen2007
andsimilarlyforthey-andz-components,wend
ǫhf|
ˆp|ii=
im
￿
ǫhf|[H
0
,
ˆ
r]|ii
=im

E
f
−E
i
￿

ǫhf|
ˆr|ii
=imω
fi
ǫhf|
ˆr|ii.
(3.49)
Thustheproblemreducestocalculatingthematrixelementshf|
ˆr|ii,hence
thetermelectricdipoleapproximation.
Inordertoshowthatinfactmostofthesedipolematrixelementsvanish,let
usresorttoanexplicitcalculation:Thewavefunctionsofhydrogenicatoms
areangularmomentumeigenstatesoftheform
8
φ
nlm
(r)=R
nl
(r)Y
m
l
(θ,φ).(3.50)
Bydenotingthequantumnumbersoftheinitialandnalstatesasn
i
,l
i
,m
i
andn
f
,l
f
,m
f
,correspondingly,weget
ǫhf|
ˆ
r|ii=
Z

0
r
2
dr
Z
dΩR
∗n
f
l
f
(r)Y
m
f

l
f
(θ,φ)ǫrR
n
i
l
i
(r)Y
m
i
l
i
(θ,φ)
=
Z

0
drr
3
R
∗n
f
l
f
(r)R
n
i
l
i
(r)
Z
dΩY
m
f

l
f
(θ,φ)ǫr
0
Y
m
i
l
i
(θ,φ),
(3.51)
wherer
0
=r/|r|.Letusconcentrateontheangularintegral.Inspherical
coordinateswehave
ǫr
0

x
sinθcosφ+ǫ
y
sinθsinφ+ǫ
z
cosθ.(3.52)
Letustrytoexpressalsothisintermsofsphericalharmonics.Recallingthat
Y
0
1
(θ,φ)=
r
3

cosθ,Y
±1
1
(θ,φ)=∓
r
3

sinθe
±iφ
,(3.53)
wend
ǫr
0
=
r

3

z
Y
0
1
+
−ǫ
x
+iǫ
y

2
Y
1
1
+
ǫ
x
+iǫ
y

2
Y
−1
1

.(3.54)
8
RecallfromKvanttiIthattheradialfunctionsdonotdependonthemagneticquantumnumberm.
60
TFY-44.130KvanttimekaniikkaII
S.Virtanen2007
Theangularintegralthuscontainsfactorsoftheform
Z
dΩY
m
f

l
f
(θ,φ)Y
m
1
(θ,φ)Y
m
i
l
i
(θ,φ).(3.55)
Since
Z

0
dφe
−im
f
φ
e
imφ
e
im
i
φ
=2πδ
m,m
f
−m
i
,(3.56)
theangularintegralcanbenonzeroonlyif
Δm=m
f
−m
i
=m=1,0,−1.(3.57)
Thisistherstselectionruleforelectricdipoletransitionssuchselection
ruleshavetobeobeyedinordertohaveanonvanishingtransition
probability.Inordertoderiveasecondselectionrulebasedonorthogonality
ofthesphericalharmonics,weusetheso-calledadditiontheoremfor
sphericalharmonics:Onecanshowthat
Y
m
1
l
1
(θ,φ)Y
m
2
l
2
(θ,φ)=
l
1
+l
2
X
l=|l
1
−l
2
|
C(l
1
l
2
m
1
m
2
;l,m
1
+m
2
)Y
m
1
+m
2
l
(θ,φ),
(3.58)
whereC(l
1
l
2
m
1
m
2
;lm)aretheClebsch-Gordancoefcientsbetweenthe
coupledanduncoupledangularmomentumbases(recallKvanttiI...).By
usingthisresult,wend
Z
dΩY
m
f

l
f
(θ,φ)Y
m
1
(θ,φ)Y
m
i
l
i
(θ,φ)
=
Z
dΩY
m
f

l
f
(θ,φ)
l
i
+1
X
l=|l
i
−1|
C(l
i
,1,m
i
,m;l,m
i
+m)Y
m
i
+m
l
(θ,φ),(3.59)
whichvanishesduetoorthogonalityofsphericalharmonicsunless
l
f
=|l
i
−1|,l
i
,l
i
+1.(3.60)
Thisisthesecondselectionruleforelectricdipoletransitions.Infact,a
furtherconstraintcomesfromparityconservation:RecallfromKvanttiIthat
φ
nlm
(−r)=(−1)
l
φ
nlm
(r),(3.61)
61
TFY-44.130KvanttimekaniikkaII
S.Virtanen2007
i.e.theunperturbedeigenstateshaveparity(−1)
l
.Nowbecauserisodd
underparityreectionsr→−r,theparityofφ
n
f
l
f
m
f
(r)rφ
n
i
l
i
m
i
(r)is
(−1)
l
f
+l
i
−1
.Unlessthisfunctionhasevenparity,i.e.
Δl=l
f
−l
i
isanoddinteger,(3.62)
thematrixelementhf|r|iivanishes.Altogetherwendtheselectionrules
Δm=0,±1,Δl=±1
(3.63)
forelectricdipoletransitions.
Notethatifagiventransitionisforbiddenonthebasisoftheseselection
rules,itmaybeallowedasahigher-ordertransitionmagneticdipole,
electricquadrupoleetc.transitionshavetheirown,somewhatdifferent
selectionrules.Anyway,sincethehigher-ordertransitionsaremuchweaker,
transitionsthatbreaktheelectricdipoleselectionrulesareingeneralmuch
slowerthanthosethatobeythem.
3.3.2Nonmonochromaticradiation
Inpractice,theradiationwhichstrikestheatomisatleastslightly
non-monochromatic,andweshouldintegratetheperturbationHamiltonian
(3.40)overallwavevectorskandsumoverpolarizationsǫ,with
A
0
=A
0
(k,ǫ),andusetheresultingperturbationtocalculatethetransition
probabilities.Duetothesquareofthematrixelement,theresulting
expressionwouldcontaincrosstermsbetweendifferentwavevector
componentsoftheradiation.However,iftheradiationisincoherent,the
crosstermswillaveragetozeroandthecorrectresultisobtainedjustby
integratingtheresult(3.43)overthedifferentwavevectors.Inthedipole
approximationthedependenceonthedirectionofkvanishes,anditis
sufcienttointegrateoverthefrequencyω;recallthatω=ck.Ifwedenote
theenergydensityoftheradiationeldperunitfrequencyintervalby

E

,
wendbyusingEq.(3.43)fortheabsorptionprobabilityfroman
62
TFY-44.130KvanttimekaniikkaII
S.Virtanen2007
inhomogeneousradiationeldtheresult
P
abs
(t)=
Z
2πq
2
m
2
￿
2
ω
2
|hf|ǫ
ˆ
p|ii|
2
"
sin

1
2
(ω−ω
fi
)t

1
2
(ω−ω
fi
)
#
2

E


(3.49)
=
2πq
2
ω
2fi
￿
2
|hf|ǫ
ˆr|ii|
2
Z
1
ω
2
"
sin

1
2
(ω−ω
fi
)t

1
2
(ω−ω
fi
)
#
2

E


(3.64)
withintheelectricdipoleapproximation.Iftheradiationenergydensityisa
smoothfunctionofω,forωt≫1wendtogoodapproximation
P
abs
(t)=
2πq
2
ω
2
fi
￿
2
|hf|ǫ
ˆr|ii|
2
1
ω
2
fi

E





ω
fi
Z
"
sin

1
2
(ω−ω
fi
)t

1
2
(ω−ω
fi
)
#
2

=

2
q
2
t
￿
2
|hf|ǫ
ˆr|ii|
2

E





ω
fi
,
(3.65)
implyingthetransitionrate
R
abs
=

2
q
2
￿
2
|hf|ǫ
ˆr|ii|
2

E





ω
fi
.(3.66)
Ifinadditiontheradiationisunpolarized,oneshouldalsotaketheaverage
oftheresultoverallpolarizationdirections:Since
9
|hf|ǫ
ˆr|ii|
2
=
1
3

|hf|ˆx|ii|
2
+|hf|ˆy|ii|
2
+|hf|ˆz|ii|
2

=
1
3
|hf|
ˆr|ii|
2
,
(3.67)
wendforunpolarizedradiationtheabsorptionrate
R
abs
=

2
q
2
3￿
2
|hf|
ˆr|ii|
2

E





ω
fi
.(3.68)
NotethatthisresultanditsderivationisverymuchanalogoustoFermi's
goldenruleinfact,itcanbederivedbyquantizingtheEMeldandby
notingthatthenalstateconsistingofhatomicstate+photonstatesiformsa
spectralcontinuum.
9
Notethatinthelaststephf|
ˆ
r|iiisanordinary(butcomplexvalued)3Dvectorand|hf|
ˆr|ii|
2
itslengthsquared.In
addition,bydenition
|hf|ǫ
ˆr|ii|
2
=
1

R
dΩ|hf|ǫ
ˆr|ii|
2
,wheretheintegralistakenoveralldirectionsofvectorǫ.One
canverifythatthisgivesjusttheaverageoverorthogonaldirections.
63