QMGreensite_merged

288 CHAPTER18. TIME-DEPENDENTPERTURBATIONTHEORY

or,using

|ψn(t)〉 = e−iωnt|φn〉

〈ψn(t)| = eiωnt〈φn| (18.19)

wehaveingeneral

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

ei(ωk−ωn^1 )t^1 ei(ωn^1 −ωn^2 )t^2 ...ei(ωnm−^1 −ωl)tm×

×〈φk|V(t 1 )|φn 1 〉〈φn 1 |V(t 2 )|φn 2 〉〈φn 2 |V(t 3 )|φn 3 〉...〈φnm− 1 |V(tm)|φl〉

(18.20)

Inwhatfollows,wereallyonlyneedthecoefficientsck(t)tofirstorderinλ:

c

(1)

k (t)=

1

i ̄h

∫t

−∞

dt′ei(ωk−ωl)t

′

〈φk|V(t′)|φl〉 (18.21)

Itisalsooftenthecasethattheperturbingpotentialfactorizesintotimeandspace-
dependentpieces,

λV(x,t)=λv(x)f(t) (18.22)

sothattofirstorder

ck(t)=δkl+λc(1)k (t)=δkl+λ〈φk|v|φl〉

1

i ̄h

∫t

−∞

dt′ei(ωk−ωl)t

′

f(t′) (18.23)

Then the transition probability Pl→k for the system, initially instate ψl at time
t→−∞,tobefoundinadifferentstateψk,k+=lattimet,isjust

Pl→k =

∣∣

∣〈ψk(t)|ψ(t)〉

∣∣

∣

2

=

∣∣

∣ck(t)

∣∣

∣

2

= λ^2

1

̄h^2

∣∣

∣〈φk|v|φl〉

∣∣

∣

2 ∣∣∣

∣

∫t

−∞

dt′ei(ωk−ωl)t

′

f(t′)

∣∣

∣∣

2

(18.24)

18.1 Harmonic Perturbations

Asalreadymentioned,theelectricpotentialassociatedwithelectromagneticradiation
variesharmonicallyintime. Letusthenconsideratime-dependentpotentialofthe
form

λV(x,t)=

{

0 t≤ 0

λv(r)2cos(ωt) t> 0

(18.25)