64 CHAPTER5. DYNAMICSOFTHEQUANTUMSTATE
ThenrewritingtheSchrodingerequation(anditscomplexconjugate)as
∂ψ
∂t=
i ̄h
2 m∂^2 ψ
∂x^2−
i
̄hVψ∂ψ∗
∂t= −
i ̄h
2 m∂^2 ψ∗
∂x^2+
i
h ̄Vψ∗ (5.12)andinsertinginto(5.11),wehave<p> = m∫
dx{(
−i ̄h
2 m∂^2 ψ∗
∂x^2+
i
̄hVψ∗)
xψ+ ψ∗x(
i ̄h
2 m∂^2 ψ
∂x^2−
i
h ̄Vψ)}= m(
i ̄h
2 m)∫
dx{−(∂^2
∂x^2ψ∗)xψ+ψ∗x∂^2
∂x^2ψ}=
∫
dxψ∗(
−i ̄h∂
∂x)
ψ (5.13)whichisthesameruleforcomputing
asinthepreviousV = 0 case.Then
∂t<p> = −i ̄h∫
dx[
∂ψ∗
∂t∂ψ
∂x+ψ∗∂^2 ψ
∂t∂x]= −i ̄h∫
dx[
∂ψ∗
∂t∂ψ
∂x−
∂ψ∗
∂x∂ψ
∂t]= −i ̄h∫
dx{(
−i ̄h
2 m∂^2 ψ∗
∂x^2+
i
̄hVψ)
∂ψ
∂x−∂ψ∗
∂x(
i ̄h
2 m∂^2 ψ
∂x^2−
i
̄hVψ)}= −
h ̄^2
2 m∫
dx[
∂^2 ψ∗
∂x^2∂ψ
∂x+
∂ψ∗
∂x∂^2 ψ
∂x^2]+
∫
dx[
ψ∗V∂ψ
∂x+
∂ψ∗
∂xVψ]
(5.14)Againapplyingintegrationbypartstothe firsttermofthefirst integral, andthe
secondtermofthesecondintegral,wefind
∂t<p> = −̄h^2
2 m∫
dx[
−∂ψ∗
∂x∂^2 ψ
∂x^2+
∂ψ∗
∂x∂^2 ψ
∂x^2]+
∫
dx[
ψ∗V∂ψ
∂x−ψ∗∂
∂x(Vψ)]=
∫
dx[
ψ∗V∂ψ
∂x−ψ∗V∂ψ
∂x−ψ∗∂V
∂xψ]