64 CHAPTER5. DYNAMICSOFTHEQUANTUMSTATE
ThenrewritingtheSchrodingerequation(anditscomplexconjugate)as
∂ψ
∂t
=
i ̄h
2 m
∂^2 ψ
∂x^2
−
i
̄h
Vψ
∂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
̄h
Vψ∗
)
xψ
+ ψ∗x
(
i ̄h
2 m
∂^2 ψ
∂x^2
−
i
h ̄
Vψ
)}
= m
(
i ̄h
2 m
)∫
dx{−(
∂^2
∂x^2
ψ∗)xψ+ψ∗x
∂^2
∂x^2
ψ}
=
∫
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
̄h
Vψ
)
∂ψ
∂x
−
∂ψ∗
∂x
(
i ̄h
2 m
∂^2 ψ
∂x^2
−
i
̄h
Vψ
)}
= −
h ̄^2
2 m
∫
dx
[
∂^2 ψ∗
∂x^2
∂ψ
∂x
+
∂ψ∗
∂x
∂^2 ψ
∂x^2
]
+
∫
dx
[
ψ∗V
∂ψ
∂x
+
∂ψ∗
∂x
Vψ
]
(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
∂x
ψ
]