QMGreensite_merged

287

i ̄h

dc(1)k

dt

=

∑

n

〈ψk|V|ψn〉c(0)n

i ̄h

dc(2)k

dt

=

∑

n

〈ψk|V|ψn〉c(1)n

... = ...

i ̄h

dc(km+1)

dt

=

∑

n

〈ψk|V|ψn〉c(nm)

... = ... (18.11)

Thissetcanbesolvediteratively,startingfromknowledgeoftheset{c(0)n }.
Supposethatinitially(sayatt→−∞),beforetheperturbationisturnedon,the
systemwasinanenergyeigenstateoftheunperturbedHamiltonianφl.Forexample,
inthecasewehaveinmind,perhapstheelectronof aHydrogenatomisinsome
definiteorbitalbeforealaser,aimedattheatom,isturnedon.Thenwehave

cn(t→−∞)=c(0)n =δnl (18.12)

Substitutingintotheequationfordc
(1)
k /dt,

i ̄h

dc(1)k

dt

=〈ψk|V|ψl〉 (18.13)

whichisintegratedtoyield

c(1)k (t)=

1

i ̄h

∫t

−∞

dt 1 〈ψk(t 1 )|V(t 1 )|ψl(t 1 )〉 (18.14)

Likewise,

ih ̄

dc(2)k

dt

=

∑

n

〈ψk|V|ψl〉c(1)n (18.15)

isintegratedto

c(2)k (t)=

1

i ̄h

∑

n 1

∫t

−∞

dt 1 〈ψk(t 1 )|V(t 1 )|ψn 1 (t 1 )〉c(1)n 1 (t 1 ) (18.16)

inserting(18.14)

c(2)k (t)=

( 1

i ̄h

) (^2) ∑
n 1
∫t
−∞
dt 1 〈ψk(t 1 )|V(t 1 )|ψn 1 (t 1 )〉
∫t 1
−∞
dt 2 〈ψn 1 (t 2 )|V(t 2 )|ψl(t 2 )〉
(18.17)
Byinduction,itsnothardtoseethat
c
(m)
k (t) =

( 1

i ̄h

)m ∑

n 1 ,n 2 ,...,nm− 1

∫t

−∞

dt 1

∫t 1

−∞

dt 2

∫t 2

−∞

dt 3 ...

∫tm− 1

−∞

dtm〈ψk(t 1 )|V(t 1 )|ψn 1 (t 1 )〉×

× 〈ψn 1 (t 2 )|V(t 2 )|ψn 2 (t 2 )〉〈ψn 2 (t 3 )|V(t 3 )|ψn 3 (t 3 )〉...〈ψnm− 1 (tm)|V(tm)|ψl(tm)〉

(18.18)