QMGreensite_merged

18.1. HARMONICPERTURBATIONS 289

Againweimaginethatthesystemisinthestateψlatt≤0,sothatck(0)=c^0 k=δkl.
Substituting(18.25)into(18.23)givesus,fork+=l,

ck(t) = λc(1)k (t)

= λ〈φk|v|φl〉

1

i ̄h

∫t

0

dt′ei(ωk−ωl)t

′[

eiωt

′

+e−iωt

′]

= −λ〈φk|v|φl〉

1

i ̄h

[

ei(ωkl−ω)t− 1

ωkl−ω

+

ei(ωkl+ω)t− 1

ωkl+ω

]

= − 2 iλ〈φk|v|φl〉

1

̄h

[

ei(ωkl−ω)t/^2

ωkl−ω

sin[

1

2

(ωkl−ω)t]+

+

ei(ωkl+ω)t/^2

ωkl+ω

sin[

1

2

(ωkl+ω)t]

]

(18.26)

wherewehavedefined

ωkl=ωk−ωl=

Ek−El

̄h

(18.27)

Clearly, the transition probability Plk = |ck|^2 is going tobe largestwhen the
angularfrequencyωoftheperturbingpotentialisequaltoeitherωklor−ωkl,i.e

h ̄ω=Ek−El “absorbtion” (Ek>El)

h ̄ω=El−Ek “stimulatedemission” (El>Ek)

(18.28)

Theterms“absorbtion”and“stimulatedemission”refertoatomicelectronsinthe
fieldofanelectromagneticwave. AccordingtotheEinstein-Bohrpicture,theincom-
ingradiationconsistsofphotons,eachofenergyE=hf= ̄hω.Anelectron,initially
inanorbitalwithenergyEl,absorbsaphotonofenergy ̄hωfromtheincomingelec-
tromagneticradiation,andjumpsintoahigherenergystateEk=El+ ̄hω. Thisis
theabsorbtioncase.
Butfromour result(18.25)wedraw some conclusionswhichcannot easily be
anticipatedfromtheBohrmodel.Thefirstsurpriseisthat,inadditiontoabsorbtion,
wheretheelectrongobblesupaphotonandjumpstoahigherenergy,thereisalso
thephenomenonofstimulatedemission,inwhichtheincomingradiation“stimulates”
anelectroninanexcitedstatewithenergyEltojumpdowntoalowerenergystate
withenergyEk. Inthiscasetheelectrondoesnotabsorbaphoton(infactitemits
one,althoughwecannotdemonstratethathere).ThetwocasesareillustratedinFig.
18.1. Thereisnosimpleclassicalpictureofstimulatedemission;itisanintrinically
quantum-mechanicalprocess. Nevertheless,thefactthatsuchaprocessisrequired
was deduced byAlbert Einsteinlongbefore thedevelopment ofmodernquantum
mechanicsbasedontheSchrodingerequation.Einstein’sreasoningwasbasedonthe
necessitiesof thermalequilibrium: ifabsorbtionispossible, stimulatedemissionis
alsorequired. Quantumtheorymustbeconsistentwithstatistical mechanics, and
happilythemysteriousstimulatedemissionturnsouttobeapredictionofthetheory.