QMGreensite_merged

296 CHAPTER18. TIME-DEPENDENTPERTURBATIONTHEORY

weget

ψ(x,t)=e−iωlt



φl(x)+λ

∑

k(=l

〈φk|V(t)|φl〉

E(0)l −Ek^0

φk(x)



 (18.64)

Noticethattheexpressioninsidethe[...]bracketsisthek-theigenstate,tofirst
orderinλ,oftheHamiltonian

H=H 0 +V′(x) (18.65)

where
V′(x)=λV(x,t) (18.66)

Likewise,theenergyexpectationvalueattimet,givenby

E(t) = 〈ψ(t)|H 0 +λV(t)|ψ(t)〉

= E^0 l+λ〈φl|V(t)|φl〉+

{

〈φl|H 0

∑

k(=l

〈φk|V(t)|φl〉

E

(0)

l −E

0

k

|φk〉+

+hermitianconjugate

}

= E^0 l+λ〈φl|V(t)|φl〉+

{∑

k(=l

〈φl|H 0 |φk〉

〈φk|V(t)|φl〉

El(0)−Ek^0

+hermitianconjugate

}

= E^0 l+λ〈φl|V(t)|φl〉 (18.67)

Werecognizethisasthesameresultasintime-independentperturbationtheory,for
aperturbingpotentialV′(x)=λV(x,t)
Tosumitup,forveryslowlyvarying(“adiabatic”)potentials,theprescriptionis
simple:Firstsolvethetime-independentSchrodingerequation

[H 0 +λV(x,t)]φl(x,t)=El(t)φl(x,t) (18.68)

wherethetimevariabletintheaboveequationisjusttreatedasafixedparameter,
subjecttothecondition
lim
λ→ 0

φl(x,t)=φl(t) (18.69)

Tofirstorderinλ,theresultis

φl(x,t) = φl(x)+λ

∑

k(=l

〈φk|V(t)|φl〉

El(0)−Ek^0

φk(x)

El(t) = El^0 +λ〈φl|V(t)|φl〉 (18.70)

and
ψ(x,t)=φl(x,t)e−iEl(t)/ ̄h (18.71)