QMGreensite_merged

18.1. HARMONICPERTURBATIONS 293

andnearlythesamematrixelements

〈φk′|v|φl〉≈〈φk|v|φl〉 (k′,k∈K) (18.43)

Forexample,theelectronfinalstatemightlieina(conduction)bandofenergies. Or,
forasingleatom,wemighttakeintoaccountthepossibilitythatsincetheenergyofan
excitedorbitalisnotprecise,thereisacontinuumofpossibleenergiesfortheemitted
photon. Anotherpossibility isthescattering ofan electronby apotential, where
thereisacontinuumofpossiblefinalmomenta%pwiththesameenergyE=p^2 / 2 m.
Inthissituation,wemaybeinterestedinthetransitionprobabilitytoanyoneofa
setoffinalstatesKintheenergyrange[Ek,Ek+∆].
Letusdefinethedensityoffinalstates

g(E)δE≡no.ofstateswithenergiesintherange[E,E+δE] (18.44)

Then the transition probability to some member of thisset of final states is the
integral

PlK(t)=λ^2

∫Ek+∆

Ek

dEk′g(Ek′)

∣∣

∣〈φk′|v|φl〉

∣∣

∣

(^24)
h ̄^2
sin^212 (ωk′l−ω)t
(ωk′l−ω)^2

(18.45)

Forlarget,theintegrandisverystronglypeakednearEk′ =El+ ̄hω,andwecan
assumethatthematrixelementofv(x),andthedensityoffinalstatesg(Ek′)isnearly
constantwithintheverynarrowrangeofenergiesinthepeak.Thenwehave

PlK(t)=λ^2 g(Ek)

∣∣

∣〈φk|v|φl〉

∣∣

∣

24

̄h^2

∫Ek+∆

Ek

dEk′

sin^212 (ωk′l±ω)t

(ωk′l±ω)^2

(18.46)

Thequickwaytodotheintegralistousethelarge-timelimit(18.40)

lim

t→∞

sin^212 (ωkl±ω)t

(ωkl±ω)^2

=

1

2

πtδ(ω±ωkl) (18.47)

andwefindthat

PlK(t)=t

2 π

h ̄

g(Ek)

∣∣

∣〈φk|v|φl〉

∣∣

∣

2

(18.48)

Thetransitionrateisdefinedastheprobabilityoftransitionperunittime.There-
fore,differentiating(18.48)wrtt,wederive

Fermi’sGolden Rule

Thetransitionratetosomememberofasetof“energy-conserving”stateswith
Ek≈El+ ̄hωisgiven,aftersufficienttime,by

Γlk=

2 π

̄h

g(Ek)

∣∣

∣〈φk|v|φl〉

∣∣

∣

2

(18.49)