25.2. THEQUANTIZATIONOFLIGHT 391
4.ThecreationoperatorandtheHamiltoniansatisfythecommutationrelation
[H,a†(k,λ)]=|k|a†(k,λ) (25.64)Problem: Verifythesestatements.
Makinguseofitems(2)and(4)above,itfollowsthattheexcitedenergyeigenstates
HΨnanb...np=Enanb...npΨnanb...np (25.65)ofthequantizedelectromagneticfieldhavetheform
Ψnanb...np=(a†(ka,λ))na(a†(kb,λ))nb...(a†(kp,λ))npΨ 0 (25.66)withenergyeigenvalues
Enanb...np=na|ka|+nb|kb|+...+np|kp|+E 0 (25.67)Restoringthefactorsof ̄handthespeedoflightc,thisexpressionisreally
Enanb...np = na ̄hc|ka|+nb ̄hc|kb|+...+np ̄hc|kp|+E 0
= na ̄hωa+nb ̄hωb+...+np ̄hωp+E 0
= nahfa+nbhfb+...+nphfp+E 0 (25.68)wherefisthefrequencyofradiationwithawavevectorofmodulus|k|.
Inclassicalphysics,any amountof energycanbestored inan electromagnetic
fieldof agivenfrequencyf. Theenergyof thefieldisproportionalto thesquare
oftheamplitude, andtheamplitudeisproportionalto thestrengthofthe source.
Whatwelearnfromquantizingtheelectromagneticfieldisthat,asinan ordinary
harmonicoscillator,theenergy(abovethegroundstate)thatcanbestoredatany
givenfrequencycomesinmultiplesofhf,i.e.
∆E=nhf (25.69)Thisrule,thattheenergyofanelectromagneticfieldateachfrequencyisquantized,
bringsusfullcircle: itexplainsblack-bodyradiation, itexplains thephotoelectric
effect,itgivesustheformulaeofPlanckandEinstein. Butwehavelearnedquitea
bitmore: wehavelearnedwhataphotonis. Itisanexcitedvibrationalquantum
stateoftheunderlyingelectromagneticfield,muchasphononsinasolidareexcited
quantumvibrationalstatesoftheunderlyingatoms.Thereisstillmore:itisbelieved
thatalloftheelementaryparticlesareessentiallyexcitedstatesofquantizedfields.
Combinequantummechanicswithclassicalfieldtheory,andparticles-or,atleast,
excitedstateswhichcarryenergyandmomentumandseemtobehaveineveryway
likepointparticles-aretheinevitableresult.Inparticular,combinequantumtheory
withMaxwell’sequations,andoutcomephotons. The beautyandthe staggering
implicationsofthisresultcan hardlybe overstated. Itisthereasonthatmodern
elementaryparticletheoryisessentiallythetheoryofquantizedfields.