QMGreensite_merged

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)