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62 CHAPTER5. DYNAMICSOFTHEQUANTUMSTATE

Ehrenfest’sPrinciple

d

dt

<qa> = <

∂H

∂pa

>

d

dt

<pa> = −<

∂H

∂qa

> (5.2)

Inparticular,foraparticleofmassmmovinginapotentialV(x)inonedimension,

d

dt

<x> = <

p

m

>

d

dt

<p> = <−

∂V

∂x

> (5.3)

Letuscheckwhetherthese equationsaresatisfiedby theequationforDeBroglie
wavesinonedimension

i ̄h

∂ψ

∂t

=−

̄h^2

2 m

∂^2 ψ

∂x^2

(5.4)

Fromthefirstequationof(5.3),weobtainanexpressionfortheexpectationvalueof
momentum

<p> = m∂t<x>

= m∂t

∫

dxψ∗(x,t)xψ(x,t)

= m

∫

dx

{

∂ψ

∂t

∗

xψ+ψ∗x

∂ψ

∂t

}

(5.5)

Then,applyingtheequation(5.4)fordeBrogliewaves

<p> = m

(

i ̄h

2 m

)∫

dx{−(

∂^2

∂x^2

ψ∗)xψ+ψ∗x

∂^2

∂x^2

ψ} (5.6)

andusingtheintegrationbypartsformula^1
∫
dx(

∂^2

∂x^2

F)G=

∫

dxF

∂^2

∂x^2

G (5.7)

thisbecomes

<p> = m

(

i ̄h

2 m

)∫

dxψ∗{x

∂^2

∂x^2

ψ−

∂^2

∂x^2

(xψ)}

= m

(

i ̄h

2 m

)∫

dxψ∗

{

x

∂^2

∂x^2

ψ−

∂

∂x

(

ψ+x

∂

∂x

ψ

)}

=

∫

dxψ∗

(

−i ̄h

∂

∂x

)

ψ (5.8)

(^1) Weassumethatthewavefunctionψanditsderivativesvanishatx=±∞,sothereareno
boundaryterms.