7.4. THEGENERALIZEDUNCERTAINTYPRINCIPLE 113
thattheresultofsuchameasurementwillbetoleavethesysteminastateofzero
uncertaintyinbothobservables,i.e.∆A=∆B=0,andthatmeansthatthephysical
statewouldhavetobeaneigenstateofbothbothA ̃andB ̃. Now,itmayormaynot
betruethatsuchphysicalstatesexist,ingeneral,forallpossibleeigenvaluesofA ̃and
B ̃. Ifnot,thenitisnotpossibletomeasuretheobservablesAandBsimultanously.
Forexample,inthecaseofpositionxandmomentump,therearenophysicalstates
whichareeigenstatesofboth ̃xandp ̃;thisiswhyitisimpossibletomeasurebothx
andpsimultaneouslyandprecisely.
Thereisasimpletestforwhetherornottwoobservablescanbemeasuredsimul-
taneously,whichrequirestheconceptofthecommutator. TheCommutatoroftwo
operatorsisdefinedtobe
[A ̃,B ̃]≡A ̃B ̃−B ̃A ̃ (7.94)If[A, ̃B ̃]=0,thetwooperatorsaresaidtocommute. Commutatorsarecomputedby
lettingthemactonanarbitraryfunction.Forexample:
[x ̃,x ̃^2 ]f(x)=(xx^2 −x^2 x)f(x)= 0 (7.95)thereforex ̃andx ̃^2 commute,i.e.
[ ̃x,x ̃^2 ]= 0 (7.96)Similarly
[x ̃,p ̃y]f(x,y,z) =[
x(−ih ̄∂
∂y)−(−ih ̄∂
∂y)x]
f(x)= −i ̄h[
(x∂f
∂y−x∂f
∂y]= 0 (7.97)
whichmeansthat ̃xandp ̃ycommute,
[x ̃,p ̃y]= 0 (7.98)However,
[x, ̃p ̃]f(x) =[
x(−i ̄h∂
∂x)−(−i ̄h∂
∂x)x]
f(x)= −i ̄h(
x∂f
∂x−
∂
∂x(xf))= −i ̄h(
x∂f
∂x−x∂f
∂x−f)= i ̄hf (7.99)