QMGreensite_merged

280 CHAPTER17. TIME-INDEPENDENTPERTURBATIONTHEORY

17.4.3 Example - TheStarkEffect

TheStarkEffectisasplittingofspectrallinesduetothe(partial)liftingofatomic
energyleveldegeneracybyanexternalelectricfield.
Letussupposethattheelectricfieldisdirectedalongthez-axis.TheHamiltonian
oftheHydrogenatomisthen

H 0 =

(

−

̄h^2

2 m

∇^2 −

e^2

r

)

−eEzz

= H 0 +λV (17.103)

wherethistime
λ=eEz and V =z (17.104)

Sincethegroundstateenergyisnon-degenerate(onlytheφ 100 statehasthisenergy
atzero-thorder),theliftingofdegeneracyfirstoccursatn=2.Therearefourstates

atn= 2 withthesameenergyE 2 (0)

|φnlm〉=|nlm〉=| 200 〉, | 211 〉, | 210 〉, | 21 − 1 〉 (17.105)

whichspana 4 × 4 subspaceofHilbertspace. Wefirsthavetocomputethe 4 × 4 V
matrixwithmatrixelements〈 2 l 1 m 1 |z| 2 l 2 m 2 〉.
Considerthecasem 1 +=m 2 .Then

〈 2 l 1 m 1 |z| 2 l 2 m 2 〉∼

∫ 2 π

0

dφei(m^2 −m^1 )= 0 (17.106)

sincez = rcos(θ)doesn’tdependon φ. Thereforeonlytermswithm 1 =m 2 are
non-zero.Next,considerl 1 =l 2 ,wherewefind

〈 2 lm|z| 2 lm〉∼

∫

dΩ|Ylm|^2 cos(θ)= 0 (17.107)

Thisisessentiallybecause|Ylm|^2 > 0 isanevenfunctionaroundθ=π/2,whilecos(θ)
isanoddfunctionforreflectionsaroundπ/2.
Thus,theonlynon-zeromatrixelementsinthissubspaceare

〈 210 |z| 200 〉 = 〈 200 |z| 210 〉

=

∫

drr^2

∫

dΩφ 200 zφ 210

=

∫

drr^2

∫

dΩ

2

(2a 0 )^3 /^2

(

1 −

r

2 a 0

)

e−r/^2 a^0 Y 00 ×

×rcos(θ)

1

√

3 (2a 0 )^3 /^2

r

a 0

e−r/^2 a^0 Y 10 (θ,φ)

= 3 a 0 (17.108)