26 CHAPTER2. ORIGINSOFQUANTUMMECHANICS
Sincetherecanonlybeanintegernumberofphotonsnatanygivenfrequency,each
ofenergy hf, theenergyofthe fieldatthatfrequencycan onlybenhf. Planck’s
restrictiononenergiesistherebyexplainedinaverynatural,appealingway.
Exceptforonelittlething. ”Frequency” isaconceptwhichpertains towaves;
yetEinstein’s suggestionis that light iscomposed of particles. The notionthat
theenergyofeach”particle”isproportionaltothefrequencyoftheelectromagnetic
”wave”,whichinturniscomposedofsuch”particles”,seemslikeaninconsistentmix
ofquitedifferentconcepts. However,inconsistentornot,evidenceinsupportofthe
existenceofphotonscontinuedtoaccumulate,asinthecaseoftheComptoneffect.
2.3 The ComptonEffect
Consideranelectromagneticwaveincidentonanelectronatrest.Accordingtoclas-
sicalelectromagnetism,thechargedelectronwillbegintooscillateatthefrequencyof
theincidentwave,andwillthereforeradiatefurtherelectromagneticwavesatexactly
thesamefrequencyastheincidentwave. ExperimentsinvolvingX-raysincidenton
freeelectronsshowthatthisisnotthecase;theX-raysradiatedbytheelectronsare
afrequencieslowerthanthatoftheincidentX-rays. Comptonexplainedthiseffect
intermsofthescatteringbyelectronsofindividualphotons.
Accordingtospecialrelativity,therelationbetweenenergy,momentum,andmass
isgivenby
E=√
p^2 c^2 +m^2 c^4 (2.14)Forparticles atrest(p= 0),this isjustEinstein’s celebratedformulaE =mc^2.
Foraparticlemovingatthespeedoflight,suchasaphoton,therestmassm=0;
otherwisethemomentumwouldbeinfinite,sincemomentumpisrelatedtovelocity
vviatherelativisticexpression
p=mv
√
1 −v
2
c^2(2.15)
Then if,foraphoton,m= 0 andE =hf,andgiventherelationforwavesthat
v=λf,wederivearelationbetweenphotonmomentumandwavelength
p=E
c=
hf
c=
h
λ(2.16)
whereλisthewavelengthoftheelectromagneticwave;inthiscaseX-rays.
NowsupposethataphotonoftheincidentX-ray,movingalongthez-axis,strikes
an electronat rest. Thephoton isscatteredat an angleθ relative to thez-axis,
whiletheelectronisscatteredatanangleφ,asshowninFig.[2.4].If%p 1 denotesthe
momentumoftheincidentphoton,%p 2 denotesthemomentumofthescatteredphoton,
and%peisthemomentumofthescatteredelectron,thenconservationofmomentum
tellsusthat
%pe=%p 1 −%p 2 (2.17)