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.