7.3. EIGENSTATESASSTATESOFZEROUNCERTAINTY 109
(II)TheprobabilityP(λα)thatanygiveneigenvaluewillbetheresultof
themeasurementis
P(λa)=|<φα|ψ>|^2 (7.63)
Theargumentforeq. (7.63)isidenticaltothereasoningofLecture 6 concerning
energyeigenvalues.Thistime,wewillcarrytheargumentthroughinbra-ketnotation.
Theexpectationvalueisgivenby
<O>=<ψ|O|ψ> (7.64)
FromTheoremH3
|ψ>=
∑
α
cα|φα> (7.65)
Todeterminethecoefficientscα,multiplybothsidesofeq. (7.65)by<φβ|
<φβ|ψ>=
∑
α
cα<φβ|φα> (7.66)
UsingTheoremH2
<φβ|ψ>=
∑
α
cαδαβ (7.67)
therefore
cβ=<φβ|ψ> (7.68)
Thebravector<ψ|correspondingtotheket|ψ>is
<ψ|=
∑
α
c∗α<φα| (7.69)
Substitutingtheseexpressionsintotheexpectationvalue
=
[
∑
α
c∗α<φα|
]
O
∑
β
cβ|φβ>
=
∑
α
∑
β
c∗αcβ<φα|O|φβ>
=
∑
α
∑
β
c∗αcβ<φα|λα|φβ>
=
∑
α
∑
β
λαc∗αcβ<φα|φβ>
=
∑
α
∑
β
λαc∗αcβδαβ
=
∑
α
λαc∗αcα
=
∑
α
λα|<φα|ψ>|^2 (7.70)