QMGreensite_merged

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)