110 CHAPTER7. OPERATORSANDOBSERVATIONS
Finally,comparethisexpressionfor
valuefromprobabilitytheory
<O>=
∑
α
OαP[Oα] (7.71)
where{Oα}arethepossiblevaluesthat theobservablecanhave. Wehavealready
shown,inthebeginningofthissection,thattheonlyvalueswhichcanbefoundby
ameasurementaretheeigenvaluesofO ̃,i.e.
{Oα}={λα} (7.72)
thentheprobabilityisuniquelydeterminedtobe
P(λa)=|<φα|ψ>|^2 (7.73)
andthisestablishestheGeneralizedBornInterpretation.
Exercise:Writeequations(7.64)through(7.73)incomponentnotation(i.e.interms
offunctionsandintegralsoverfunctions).
Letusrecordandreviewthesolutionstotheeigenvalueequationsseenthusfar:
EigenstatesofPosition
For
x ̃=x (7.74)
theeigenvalueequationis
xφx 0 (x)=x 0 φx 0 (x) (7.75)
whichhassolutions
{eigenstates φx 0 (x)=δ(x−x 0 ), eigenvalues x 0 ∈[−∞,∞]} (7.76)
withinnerproducts
<φx 1 |φx 2 >=δ(x 1 −x 2 ) (7.77)
AsaconsequenceoftheoremH3,anyarbitraryfunctionψ(x)can bewrittenas a
superpositionoftheseeigenstates:
ψ(x) =
∫
dx 0 cx 0 φx 0 (x)
=
∫
dx 0 cx 0 δ(x−x 0 ) (7.78)
whichisseentobesatisfiedbychoosing
cx 0 =ψ(x 0 ) (7.79)