18.1. HARMONICPERTURBATIONS 291
ofthewaveisct, thenthephotonislocalizedwithinthisregionbeforeabsorbtion.
Therefore,bytheUncertaintyPrinciple,thephotonhasanuncertaintyinmomentum
∆p∼h
∆x=
h
ct(18.35)
Usingtherelativisticrelationbetweenenergyandmomentumformasslessparticles,
E=pc,thephotonsinthebeamthereforehaveanuncertaintyinenergyoforder
∆E=
2 πh ̄
t(18.36)
Butthisisthesamemagnitudeastheapparent“violation”ofenergyconservationin
(18.31).Weconcludethatthereisnoviolationatall,andthatthefailureoftheBohr
relationtoholdexactly isjustaconsequenceofthe UncertaintyPrincipleapplied
tophotons,whichtellsusthattheenergyofphotonsinanelectromagneticwaveof
frequencyfisnotexactlyhf,ifthewavehasafiniteextensioninspace.
Anotherpuzzlingfactistheexistenceofspontaneousemissionofphotonsbyelec-
trons. Thisoccurswhenanelectroninanexcitedorbitalemitsaphoton,anddrops
toalowerenergyorbital,withoutbeingsubjectedtoelectromagneticradiationorany
otherexternalpotential.Anexternalradiationfieldoftherightfrequencyspeedsup
(or“stimulates”)theprocess,butemissionwilloccurinanycase,eventually. But
thisfactdirectlycontradictsthenotionthatthewavefunctionsφn(x)areeigenstates
ofenergy,andthereforestationary. Astationarystate,excitedornot,isbydefinition
independentoftime;itcannotchangeunlessactedonbyanexternalperturbation.
Asithappens,theexcitedstatesoftheHydrogenatomarenotreallyeigenstatesof
energyafterall.Non-relativisticquantummechanics,astaughtinthiscourse,isonly
anapproximationtotherealworld;inparticularitisanapproximationinwhichthe
electromagneticfieldistreatedclassically. Inthetheoryofquantumelectrodyamics,
anelectronisinconstantinteractionwiththequantizedelectromagneticfieldevenin
theabsenceofanexternalelectromagneticwave.ExcitedstatesoftheHydrogenatom
arenoteigenstatesinquantumelectrodynamics,butinsteadhavesomeuncertainty
inenergy. Fromourdiscussionofthe time-energyuncertainty relationinthefirst
semester,itisclearthatthisuncertaintyisrelatedtotheaveragetimeittakesfor
thestatetochangeinsomenoticeableway,e.g.toemitaphoton. Letuscallthis
averagetimeτ.Thentheuncertainty∆Einagivenatomicorbitalisgivenby
∆E≈
̄h
τ(18.37)
Theuncertaintyintheenergyofatomicorbitalsleadstoacorrespondingbroadening
(spreadinfrequencies)ofthephotonsemittedinelectrontransitionsbetweenthose
orbitals.
Finally, Iwantto mention briefly (and inadequately) aclever andtechnologi-
callyimportantapplicationofstimulatedemission.Thisisthelaser,whoseacronym