2.4. THEHEISENBERGMICROSCOPE 27
while,fromconservationofenergy,wehave
E 1 +mc^2 =E 2 +
√
p^2 ec^2 +m^2 c^4 (2.18)
InthisformulaE 1 istheenergyoftheincidentphoton,andmc^2 istheenergyofan
electronatrest,whileE 2 istheenergyofthescatteredphoton,and
√
p^2 ec^2 +m^2 c^4 is
theenergyofthescatteredelectron. UsingthefactthatE=pcforphotons,wehave
p^2 ec^2 +m^2 c^4 =[p 1 c−p 2 c+mc^2 ]^2 (2.19)
Squarethemomentumconservationequationtofindp^2 e
p^2 e = (%p 1 −%p 2 )·(p% 1 −%p 2 )
= p^21 +p^22 − 2 p 1 p 2 cosθ (2.20)
andinserttheexpressionforp^2 eintotheenergyconservationequation(2.19),tofind
1
p 2
−
1
p 1
=
1
mc
(1−cosθ) (2.21)
Finally,usingtherelation(2.16),Comptondeducedthat
λ 2 −λ 1 =
h
mc
(1−cosθ) (2.22)
which,infact,agreeswithexperiment.
2.4 The Heisenberg Microscope
Theparticle-likebehavioroflightwaveshassometroublingimplicationsforthecon-
ceptofaphysicalstate,asunderstoodbyclassicalphysics. Theclassicalstateofa
pointlikeparticleatanymomentintimeisgivenbyitspositionanditsmomentum
(%x,p%),whicharesupposedtobedeterminedbymeasurement.Ofcourse,thispresup-
posesthatonecanmeasurepositionandmomentumsimultaneously,toanyrequired
degreeofprecision. Buttherelationp=h/λsuggeststhatsuchsimultaneousmea-
surementsmightnotbepossible,atleast,notifsuchmeasurementsinvolvetheuse
oflight. Roughly,thereasonisthis: Inordertodetermineparticlepositiontoan
accuracy ∆x, itisnecessaryto uselightof wavelengthλ< ∆x. Butthat means
thatthephotonscomposinglightofsuchawavelengthcarrymomentump>h/∆x.
Inordertoobservetheparticle’sposition,theparticlemustscatterlight. Butthis
meansthatthescatteredphotoncanimpartmuchofitsmomentumtotheobserved
particle,changingthemomentumoftheparticlebysomeundeterminedamountof
order∆p≈h/∆x. Theproductofthetwouncertaintiesistherefore
∆x∆p≈∆x
h
∆x
=h (2.23)