7.4. THEGENERALIZEDUNCERTAINTYPRINCIPLE 115
wherebissomeconstant.Inthatcase,fromthedefinitionofφ′,
B ̃φa(x)=bφa(x) (7.107)
Thisprovesthateveryeigenstateof A ̃isaneigenstateofB ̃. Suchstatesarezero-
uncertaintystatesofbothAandB,sosimultaneousmeasurementispossible.
Nextweprovethe”onlyif”partofthecommutatortheorem.Supposetheoutcome
ofameasurementisthat A=aandB =b. Sincethe eigenvaluesof A ̃arenon-
degenerate,theresultingphysicalstateisφa,where
A ̃φa=aφa (7.108)
However,thismustalsobeazero-uncertaintystateforB,andtherefore
B ̃φa=bφa (7.109)
Itfollowsthat
AB|φa> = bA|φa>
= ba|φa> (7.110)
while
BA|φa> = aB|φa>
= ab|φa> (7.111)
therefore
[A,B]|φa> = (AB−BA)|φa>
= (ba−ab)|φa>
= 0 (7.112)
Now,accordingtotheoremH3,anysquare-integrablefunctioncanberepresentedas
alinearcombinationofeigenstatesofA ̃,i.e.
|f>=
∑
a
ca|φa> (7.113)
andthismeansthat
[A, ̃B ̃]|f> =
∑
a
ca[A, ̃B ̃]|φa>
= 0 (7.114)
Iftheoperator[A ̃,B ̃]actingonanyfunctionf(x)gives0,thentheoperatoritselfis
zero
[A, ̃B ̃]= 0 (7.115)