98 CHAPTER7. OPERATORSANDOBSERVATIONS
=
∫
dxψ∗(x,t)H ̃ψ(x,t) (7.1)
where
̃xψ(x,t) ≡ xψ(x,t)
p ̃ψ(x,t) ≡ −i ̄h
∂
∂x
ψ(x,t)
H ̃ψ(x,t) ≡
{
−
̄h^2
2 m
∂^2
∂x^2
+V(x)
}
ψ(x,t) (7.2)
Theprobabilitiesforfindingaparticlenearacertainpositionx 0 ,ormomentump 0 ,
oratacertainenergyEn,aregivenby
Pdx(x 0 ) = |ψ(x 0 )|^2 dx
Pdp(p 0 ) = |
1
√
2 π ̄h
f(p 0 )|^2 dp
P(En) = |an|^2 (7.3)
where
an = <φn|ψ>
f(p 0 ) =
∫
dxφ(x)e−ip^0 x/ ̄h (7.4)
Aprobabilityisanumber,aquantumstate|ψ>isavector. Numberscan be
obtainedfromvectorsbytakinginnerproducts. TheprobabilityP(En)isclearlythe
squaremodulusofaninnerproduct,andinfacttheothertwoprobabilitiescanalso
beexpressedinthatway. Define
φx 0 (x) ≡ δ(x−x 0 )
φp 0 (x) ≡
1
√
2 π ̄h
eip^0 x/ ̄h (7.5)
Thenwemaywrite
ψ(x 0 ) =
∫
dxδ(x−x 0 )ψ(x)
= <φx 0 |ψ> (7.6)
andalso
1
√
2 π ̄h
f(p) =
∫
dx
[
1
√
2 π ̄h
eip^0 x/ ̄h
]∗
ψ(x,t)
= <φp 0 |ψ> (7.7)