QMGreensite_merged

6.1. THEEXPECTATIONVALUEOFPN 87

Then,usingthedeltafunctiontoeliminateoneofthep-integrations,wefind

<p>=

∫

dpp

f∗(p)f(p)

2 π ̄h

(6.9)

Comparisonofthisexpressiontothecorrespondingexpressionforposition

<x>=

∫

xPdx(x) (6.10)

showsthattheprobabilityPdp(p)tofindtheparticlemomentuminaninfinitesmal
rangedparoundmomentumpis

Pdp(p)=

f∗(p)f(p)

2 π ̄h

dp (6.11)

Thentheexpectationvalue<G(p)>ofanyfunctionofmomentumisgivenby

<G(p)>=

∫ dp

2 π ̄h

G(p)f∗(p)f(p) (6.12)

andinparticular

<pn>=

∫ dp

2 π ̄h

pnf∗(p)f(p) (6.13)

Itisusefultoexpressdirectlyintermsofthewavefunctionφ(x),rather
thanitsFouriertransform. Usingeq. (5.54)toreexpressf(p)intermsofφ(x),and
insertingthisexpressioninto(6.13),wehave

<pn> =

∫ dp

2 π ̄h

pn

{∫

dx 1 φ(x 1 )e−ipx^1 / ̄h

}∗∫

dx 2 φ(x 2 )e−ipx^2 / ̄h

=

∫

dx 1 φ∗(x 1 )

∫

dx 2 φ(x 2 )

∫ dp

2 π ̄h

pnei(x^1 −x^2 )p/ ̄h

=

∫

dx 1 φ∗(x 1 )

∫

dx 2 φ(x 2 )

(

i ̄h

∂

∂x 2

)n∫

dp

2 π ̄h

ei(x^1 −x^2 )p/ ̄h

=

∫

dx 1 φ∗(x 1 )

∫

dx 2 φ(x 2 )

(

i ̄h

∂

∂x 2

)n

δ(x 1 −x 2 ) (6.14)

Integratingbypartsntimeswithrespecttox 2 ,thisbecomes

<pn> =

∫

dx 1

∫

dx 2 φ∗(x 1 )δ(x 1 −x 2 )

(

−i ̄h

∂

∂x 2

)n

φ(x 2 )

=

∫

dxφ∗(x)

(

−i ̄h

∂

∂x

)n

φ(x) (6.15)

Theconclusionisthattheexpectationvalueatanytimetisjust

<pn>=

∫

dxψ∗(x,t)p ̃nψ(x,t) (6.16)

wherep ̃isthemomentumoperator

p ̃=−i ̄h

∂

∂x

(6.17)

introducedinthelastlecture.