114 CHAPTER7. OPERATORSANDOBSERVATIONS
Fromthiswecanconclude thattheoperatorscorrespondingto positionalongthe
x-axis,andmomentuminthex-direction,donotcommute,andthat
[x ̃,p ̃]=i ̄h (7.100)Problem:Computethecommutators:[H ̃,x ̃],[H ̃,p ̃],[p ̃^2 , ̃x^2 ].
Theconditionthattwoobservablescanbemeasuredsimultaneouslyisthenstated
by:
TheCommutatorTheorem
Two observables A and B are simultaneously measureable (have the
same setof eigenstates) ifand only their corresponding operatorscom-
mute,i.e. [A, ̃B ̃]= 0.
Thefact that positionand momentum arenot simultaneously observableis a
specialcaseofthistheorem. Forsimplicity,wewillprovetheoremonlyinthecase
thattheeigenvaluesofA ̃andB ̃arediscreteandnon-degenerate.
Letusbeginwiththe”if”partofthetheorem:[A, ̃B ̃]= 0 impliesthateigenstates
ofA ̃areeigenstatesofB ̃andviceversa. Now[A, ̃B ̃]= 0 meansthat
A ̃B ̃f(x)=B ̃Af ̃ (x) (7.101)andletφabeanyeigenstateofA ̃,i.e.
A ̃φa=aφa (7.102)Then
A ̃B ̃φa(x) = B ̃A ̃φa(x)
= aB ̃φa(x) (7.103)Nowifwedefine
φ′a=B ̃φa (7.104)
theneq.(7.103)becomes
A ̃φ′a=aφ′a (7.105)
whichmeansthatφ′aisaneigenstateofA ̃witheigenvaluea.Butsincetheeigenvalues
ofA ̃arenon-degenerate,itcanonlybethatφ′aisproportionaltoφa,i.e.
φ′a(x)=bφa(x) (7.106)