QMGreensite_merged

286 CHAPTER18. TIME-DEPENDENTPERTURBATIONTHEORY

agivenorbitalφn,whatistheprobabilitythattheincidentradiationwillresultin
theelectronbeingfoundintheorbitalφmatsomelatertimet?
Tobeginwith,sincethestates{ψn(x,t)}spantheHilbertspaceatanytimet,we
canalwaysexpandthesolutionofthetime-dependentSchrodingerequationψ(x,t),
atanytimet,inthatbasis:

ψ(x,t)=

∑

n

cn(t)ψn(x,t) (18.4)

Theinitialvalueproblem,i.e.“givenψ(x,0),findψ(x,t),”canbephrasedas:given
{cn(0)},find{cn(t)}.Substituting(18.4)intothetime-dependentSchrodingerequa-
tion

i ̄h

d

dt

ψ=(H 0 +λV)ψ (18.5)

toget

i ̄h

∑

n

(

dcn

dt

−iωncn

)

ψn=

∑

n

(

̄hωn+λV

)

cnψn (18.6)

Cancellingtermsoneachsideproportionalto ̄hωn,andgoingtoketnotation

i ̄h

∑

n

dcn

dt

|ψn〉=

∑

n

λVcn|ψn〉 (18.7)

Multiplybothsidesby〈ψk|

i ̄h

dck

dt

=λ

∑

n

〈ψk|V|ψn〉cn (18.8)

Obviously,thesolutionscn=cn(t)areareactuallyfunctionsofbothtimeandλ. We
canthereforeexpandthecninaTaylorseriesaroundλ= 0

ck(t) = c(0)k +λc(1)k (t)+λ^2 c(2)k (t)+...

c(kn)(t) =

1

n!

(

dnck

dλn

)

λ=0

(18.9)

Substitutingthisexpansioninto(18.8)

i ̄h



dc

(0)

k

dt

+λ

dc(1)k

dt

+λ^2

dc(2)k

dt

+...



=

∑

n

〈ψk|V|ψn〉

[

λc(0)n +λ^2 c^1 n+λ^3 c(2)n +...

]

(18.10)

andequatingequalpowersofλontherightandleft-handsidesgivesaninfiniteset
ofequations

i ̄h

dc(0)k

dt

= 0