QMGreensite_merged

110 CHAPTER7. OPERATORSANDOBSERVATIONS

Finally,comparethisexpressionforagainsttheexpressionforanexpectation
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)