Chapter 18
Time-Dependent Perturbation
Theory
ThusfarwehavealwaysassumedthatthepotentialV(x)intheSchrodingerequa-
tionistime-independent. Butnothinginthisworldiscompletelytimeindependent,
andsomeofthemostinteresting,andtechnologicallyimportant,aspectsofatomic
physicsconcerntheinteractionofatomswithelectromagneticradiation. Electromag-
neticwavesconsistoforthogonalelectricandmagneticfieldswhichoscillate,atany
point,harmonicallyintime.Ifthefieldstrengthissmallenough,thentheassociated
electrostaticpotentialcan be viewedasa small,time-varying perturbationtothe
usualHamiltonianoftheatom. Sotounderstandtheeffectofelectromagneticwaves
onatomic electrons,we needto developmethodsfordealing withtime-dependent
perturbationsoftheatomicpotential.
Letusthenconsidertime-dependentHamiltoniansoftheform
H=H 0 +λV(x,t) (18.1)whereweassumethattheeigenstatesofH 0 ,nowdenotedφn,
H 0 φn(x)=Enφn(x) (18.2)areknown.Denotethecorrespondingenergy-eigenstatesolutionsofthetime-dependent
Schrodingerequationby
ψn(x,t)=φn(x)e−iωnt ωn=En
̄h(18.3)
Justto have adefinitepictureinmind,we mightimaginethatthe φn arethe
energy eigenstatesof theHydrogen atom, andthetime-varying potentialλV(x,t)
istheelectrostaticpotential,atpointx,ofanincidentelectromagneticwave. The
questionweareinterestedinansweringisthis:Iftheelectronstartsoutattimet 0 in