QMGreensite_merged

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)