QMGreensite_merged

86 CHAPTER6. ENERGYANDUNCERTAINTY

6.1 The Expectation Value of p
n

AccordingtotheBornInterpretation,theexpectationvalueofanyfunctionF(x)of
positionisgivenby

<F(x)> =

∫

F(x)Pdx(x)

=

∫

dxF(x)ψ∗(x,t)ψ(x,t) (6.2)

whilefromEhrenfest’sPrinciple,onefindsthattheexpectationvalueofmomentum
is

<p>=

∫

dxψ∗(x)

(

−ih ̄

∂

∂x

)

ψ(x,t) (6.3)

Butwewouldalsoliketoknowhowtocomputetheexpectationvalueofenergy

<E>=<H> = <

p^2

2 m

+V(x)>

=

1

2 m

<p^2 >+<V(x)> (6.4)

aswellastheuncertaintyinmomentum

∆p^2 =<p^2 >−<p>^2 (6.5)

andtoevaluateeitherofthesequantities,itisnecessarytoevaluate<p^2 >. What
isneededisaprobabilitydistributionformomentum, Pdp(p),analogoustoPdx(x),
givingtheprobabilitythattheparticlemomentumwillbefoundwithinasmallrange
ofmomentaaroundp. Thisdistributioncanbededucedfromtheruleforcomputing

,andtheinverseFouriertransformrepresentation(5.50).
Atagiventimet 0 ,thewavefunctionisafunctionofxonly, whichwe denote,
again,by
φ(x)≡ψ(x,t 0 ) (6.6)

andexpectationvalueofmomentumatthatinstantisgivenby

<p>=

∫

dxφ∗(x)

(

−ih ̄

∂

∂x

)

φ(x) (6.7)

Expressingφ(x)intermsofitsinverseFouriertransform,eq. (5.50),andproceeding
asineq. (5.75)

<p> =

∫

dxφ∗(x)

(

−i ̄h

∂

∂x

)

φ(x)

=

∫

dx

{∫

dp 1

2 π ̄h

f(p 1 )eip^1 x/ ̄h

}∗(

−i ̄h

∂

∂x

)∫

dp 2

2 πh ̄

f(p 2 )eip^2 x/ ̄h

=

∫ dp

1

2 π ̄h

f∗(p 1 )

∫ dp

2

2 π ̄h

p 2 f(p 2 )

∫

dxei(p^2 −p^1 )x/ ̄h

=

∫ dp

1

2 π ̄h

f∗(p 1 )

∫ dp

2

2 π ̄h

p 2 f(p 2 )2πh ̄δ(p 1 −p 2 ) (6.8)