QMGreensite_merged

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)