QMGreensite_merged

266 CHAPTER17. TIME-INDEPENDENTPERTURBATIONTHEORY

Thistime

H 0 =−

̄h^2

2 m

∇^2 −

e^2

r

(17.9)

istheusualHydrogenatomHamiltonian,and

alittlebitextrapotential=eEz (17.10)

Ofcourse,ineachcasetheseextrapotentialscanonlybecharacterizedas“alittlebit
extra”iftheconstantsλandeEaresmall,insomeappropriateunits(tobediscussed
furtherbelow). Letswriteingeneralthat

alittlebitextrapotential=V′(x)=λV(x) (17.11)

whereλissome constantwhichcontrolswhetherV′(x)issmallor not. Then the
solutionsoftheeigenvalueequation(17.3)are,obviously,funtionsofλ,whichcanbe
expandedinaTaylorseriesaroundλ=0,i.e.

φn = φn(x,λ)

= φ(0)n (x)+λφ(1)n (x)+λ^2 φ(2)n (x)+...

En = En(λ)

= E(0)n +λE(1)n +λ^2 En(2)+... (17.12)

where

E(nk) ≡

1

k!

(

dk

dxk

En(λ)

)

λ=0

φkn(x) ≡

1

k!

(

dk

dxk

φ(x,λ)

)

λ=0

(17.13)

andofcourse
H=H 0 +λV (17.14)

Thentheeigenvalue(17.3)becomes

(H 0 +λV)(φ(0)n +λφ(1)n +λ^2 φ(2)n +...)

= (En(0)+λEn(1)+λ^2 En(2)+...)(φ(0)n (x)+λφ(1)n (x)+λ^2 φ(2)n (x)+...)

(17.15)

Collectinglikepowersofλoneachsidewehave

H 0 φ(0)n +λ(H 0 φ(1)n +Vφ(0)n )+...+λN(H 0 φ(nN)+Vφ(nN−1))+...

= En(0)φ(0)n +λ(En(0)φ(1)n +En(1)φ^0 n)+λ^2 (En(0)φ(2)n +E(1)n φ(1)n +En(2)φ(0)n )+...

+λN

∑N

j=0

E(nj)φ(nN−j)+... (17.16)