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