290 CHAPTER18. TIME-DEPENDENTPERTURBATIONTHEORY
ThesecondsurpriseisthatalthoughthetransitionfromEltoEkbyabsorbtionis
mostprobablewhentheincomingradiationhasanangularfrequencyω=(Ek−El)/ ̄h,
theprobabilityisnon-zeroawayfromthisvaluealso. Infact,forEk lessthanbut
closetoEl,thetransitionprobabilityasafunctionoftimeisroughly
Plk(t)=λ^2∣∣
∣〈φk|v|φl〉∣∣
∣(^24)
̄h^2
sin^212 (ωkl−ω)t
(ωkl−ω)^2
(18.29)
Fig.18.2isagraphofthefunction
g(ωkl−ω)=sin^212 (ωkl−ω)t
(ωkl−ω)^2(18.30)
atafixedtimet(ithasthesamefunctionalformastheintensityvs.distanceofa
one-slitdiffractionpattern).Thecentralpeakhasahalf-widthof 2 π/t,whichmeans
thattheexpectedrelationforenergyconservation,i.e.
Ek=El+ ̄hω (18.31)isonlysatisfiedinthet→∞limit. Foranyfinitet,thereisafiniteprobabilityfor
(18.31)tobeviolated. Itseemsasthoughwehavelostenergyconservation! Howis
thispossible?
Ofcoursethere isreally no breakdownof energy conservation. The apparent
breakdown,inthiscase,comesfromtheassumptionthattheelectromagneticradi-
ationincidenton the atomonlycontains photonsof definiteenergy ̄hω; andthis
assumptionactuallyconflictswiththeHeisenbergUncertainty Principle. Letssee
howthisworks. Firstofall,ifwewrite
ωkl=ω+∆ω (18.32)thentherewillnormallyonlybeatransitionbetweenorbitalsbyabsorbtionif∆ωis
withinthe“centralpeak”
∆ω≈±2 π
t(18.33)
sotheapparentviolationofenergyconservationisoftheorder
∆E=h ̄∆ω=2 ̄hπ
t(18.34)
Now byassumption, the perturbing potential was turnedon at t = 0, so the
portionoftheelectromagneticwave,containingthephotonabsorbedbytheelectron,
hasanextensioninspaceof∆x=ct.Theelectromagneticwaveitselfcanbethought
ofasthewavefunctionofthephoton(aconceptwhichcanonlybemadepreciseinthe
frameworkofrelativisticquantumfieldtheory),andthismeansthatiftheextension