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5.2. THESCHRODINGERWAVEEQUATION 65

=

∫

dxψ∗

(

−

∂V

∂x

)

ψ

= <−

∂V

∂x

> (5.15)

exactlyasrequiredbyEhrenfest’sprinciple.
WeseethatEhrenfest’sprincipleandtheSchrodingerequation,takentogether,
makethefollowingpredictionfortheexpectationvalueofparticlemomentumattime
t,regardlessofthepotentialV(x)

<p>=

∫

dxψ∗(x,t)p ̃ψ(x,t) (5.16)

wherep ̃isthedifferentialoperatorknownastheMomentumOperator

p ̃≡−i ̄h

∂

∂x

(5.17)

Thissupplies an importantpiece of information. The Borninterpretation of the
wavefunctiontellsushowtocomputetheexpectationvalueofofpositionofa
particle,givenitswavefunctionψ(x,t). Butpositionisnottheonlyobservableofa
particlethatcanbemeasured;forexample,onecanalsomeasureaparticle’smomen-
tum,andtheBorninterpretationdoesnottellushowtocomputethecorresponding
expectationvalue

. Nowwehavetheruleforfinding

;itiscontainedin
equations(5.16)and(5.17)above.
Eq. (5.17)canbeusedtoexpressanyx-derivativeintermsofthemomentum
operator,andinparticular

∂^2

∂x^2

=−

1

̄h^2

p ̃^2 (5.18)

ThentheSchrodingerequationcanbewritteninthecompactform

i ̄h∂tψ =

(

p ̃^2

2 m

+V

)

ψ

= H ̃ψ (5.19)

where

H[p,q]=

p^2

2 m

+V (5.20)

isjusttheHamiltonianforaparticleofmassm,movinginapotentialfieldV,and
H ̃ istheHamiltonianOperator

H ̃=H[p, ̃x] (5.21)

obtainedbyreplacingmomentumpbythemomentumoperatorp ̃intheHamiltonian
function.