QMGreensite_merged

298 CHAPTER18. TIME-DEPENDENTPERTURBATIONTHEORY

18.2.2 Validity

Beforeleavingthetopic,itsnecessarytohavesomecriterionforthevalidityofthe
adiabaticmethod:whatdowemeanbysayingthatthepotentialis“slowlyvarying”?
Slowlyvaryingcomparedtowhat?
Letsgobacktotheintegration-by-partsformula,andassumethattheperturbing
potentialwasturnedonatt= 0

c^1 k(t)=−

1

̄hωkl

[

Vkl(t)eiωklt−

∫t

0

dt′

∂Vkl

∂t′

eiωklt

′

]

(18.75)

Makingtheroughassumptionthat ∂∂Vtkl isnearlyconstantintime,wecancarryout
thetimeintegrationtoget

c^1 k(t)=−

1

̄hωkl

[

Vkl(t)eiωklt−

1

iωkl

∂Vkl

∂t

2 sin[

1

2

ωklt]eiωklt/^2

]

(18.76)

WeareonlyjustifiedindroppingthetermproportionaltothetimederivativeofVkl
if
∣∣
∣Vkl(t)

∣∣

∣ 6

∣∣

∣∣

∣

2

ωkl

∂Vkl

∂t

∣∣

∣∣

∣ (18.77)

Sothisisthenecessaryconditionforthevalidityoftheadiabaticapproximation.

18.3 Sudden Perturbations

Finally,letusconsiderthe otherextremeof time-dependentperturbation,namely,
thecasewherethepotentialchangesinstantly(ornearlyso,e.g.bysomeoneflipping
aswitch):

V′(x,t)=

{

0 t< 0

V′(x) t≥ 0

(18.78)

Wewillsupposethatthepossibleenergyeigenstatesandeigenvaluesbothbeforeand
aftert= 0 areknown:

H 0 φn = Enφn

Hφ′n = En′φ′n

H = H 0 +V′(x,t) (18.79)

Possibly{φ′n,En′}candeterminedbytime-independentperturbationtheory,orare
evenknownexactly(forsomesimpleformsofV′(x)).
Considerasystemwhichisinaninitialstate

ψin(x,t= 0 −) (18.80)