QMGreensite_merged

292 CHAPTER18. TIME-DEPENDENTPERTURBATIONTHEORY

(as you surely know) standsfor “LightAmplification by StimulatedEmissionof
Radiation”. Thesimplestversionoftheschemeinvolvesanatomwiththreeenergy
levels,ofenergiesE 2 >E 1 >E 3 ,withthepropertythatelectronswhichareexcited
(fromthegroundstateatE 0 )tolevelE 2 tendtorapidly(withinabout 10 ns)emit
aphotonandfallintotheE 1 level,whichiscomparativelylong-lived;ontheorder
ofmicro-to milli-secondsbeforeemittingaphoton. Theideawithlasers(ortheir
microwaverelatives,themasers),istoexciteagreatmanyelectronsfromtheground
stateintotheE 2 state,fromwhichtheyfallintotheE 1 orbital. Withenoughenergy
expended,itispossibletoobtaina“populationinversion”;i.e.moreelectronsinthe
excitedstateatE 1 thaninthegroundstate. Whatthenhappensisthatelectrons
fallingfromE 1 to thegroundstateemit photons,andthese photonscause other
electronstomakethetransition tothe groundstateby stimulatedemission. The
photonswhichstimulatetheemission,andtheemittedphotons,haveaboutthesame
energy(andthereforefrequency),andtheyareinphase.Thusthelightisextremely
coherent. Thisisincontrasttothelightemittedbythermalradiation(e.g.byalight
bulb),wheretherelativephaseinthewavefunctionofdifferentphotonsisrandom,the
black-bodydistributionoffrequenciesisbroad,andthelightissaidtobeincoherent.

18.1.1 Fermi’sGolden Rule

Wehaveseenthattheprobabilityofanabsorbtiontransitionisgivenbytheexpression

Plk(t)=λ^2

∣∣

∣〈φk|v|φl〉

∣∣

∣

(^24)
̄h^2
sin^212 (ωkl−ω)t
(ωkl−ω)^2

(18.38)

withasimilarexpressionforstimulatedemission(ωkl−ω→ωkl+ω).Itseemsthat
thisexpressioncouldbesimplified,inthet→∞limit,usingtheidentity

f(0)=

∫

dωf(ω)δ(ω)=lim

t→∞

∫

dωf(ω)

2

π

sin^212 ωt

tω^2

(18.39)

or,looselyspeaking

δ(ω)=lim

t→∞

2

π

sin^212 ωt

tω^2

(18.40)

Usingthisidentity, itseems that wecan expressthe transitionprobability inthe
t→∞limitas

Plk=λ^2

2 πt

̄h^2

∣∣

∣〈φk|v|φl〉

∣∣

∣

2

δ(ω±ωkl) (18.41)

Atthispointwehavetostop,becausesomethingappearstohavegonewrong. A
probabilityproportionaltoaDiracδ-functionisclearlyabreakdownofthenotion
thataperturbationissupposedtobeasmallcorrection.
Soletsbackupabit. ItisoftenthecasethatthereisagroupKoffinalstates
{φk′,k′∈K}whichhavenearlythesameenergies,i.e.

Ek′=Ek+! (k′,k∈K) (18.42)