116 CHAPTER7. OPERATORSANDOBSERVATIONS
whichprovesthe”onlyif”portionoftheorem.
Theproofaboveextendstriviallytocontinuousnon-degenerateeigenvalues,sim-
plyreplacingadiscreteindexbyacontinuous index,andsums byintegrals. The
commutatortheoremisalsotrueiftheeigenvaluesaredegenerate,butthosecompli-
cationswillbepostponedtoalaterlecture.
Thereisageneralizationof theHeisenbergUncertaintyPrincipletothecaseof
anytwoobservableswhichcannotbesimultaneouslymeasured:
TheGeneralizedUncertaintyPrinciple
Theuncertainties ∆Aand ∆B ofanytwoobservablesin any physical
state|ψ>satisfytheinequality
∆A∆B≥
1
2
|<ψ|[A,B]|ψ>| (7.116)Proof:Beginwiththeexpressionsforuncertainty
(∆A)^2 = <ψ|(A ̃−<A>)^2 |ψ>
(∆B)^2 = <ψ|(B ̃−<B>)^2 |ψ> (7.117)anddefinetheHermitianoperators
D ̃A≡A−<A> and D ̃B=B−<B> (7.118)andalso
|ψ 1 >=DA|ψ> and |ψ 2 >=DB|ψ> (7.119)
Then
(∆A)^2 = <ψ|DADA|ψ>
= <ψ|DA|ψ 1 >
= <DAψ|ψ 1 > (hermiticity)
= <ψ 1 |ψ 1 > (7.120)Similarly,
(∆B)^2 =<ψ 2 |ψ 2 > (7.121)
Now,makinguseoftheCauchy-Schwarzinequalityforvectors(seeproblembelow)
|u||v|≥|<u|v>| (7.122)