164
3 Quantum Mechanics – II
The first term on the RHS is zero at both the limits.
−
∂
2
ψ
∗
∂ x 2 xψdx = −
dψ
∗
dx
ψ + x
dψ
dx
dx
= ψ
dψ
∗
dx
dx +
x
dψ
∗
dx
dψ
dx
dx
(8)
Substituting (7) and (8) in (5), the terms underlined vanish together.
d < x >
dt
=
i
2m
−
ψ
∗ ∂
∂ x
dx +
ψ
dψ
∗
dx
dx
=
1
2m
ψ
∗
−i
∂
∂ x
ψ dx +
ψ
i
∂
∂ x
ψ
∗ dx
(9)
Now the operator for P x is −i
∂
∂ x
. The first term on RHS of (9) is the
average value of the momentum P x , the second term must represent the
average value of P
∗
x . But p x being real, P
∗
x = P x . Therefore
d < x >
dt
=
1
m
< P x >
(10)
Thus (10) is similar to classical equation x = p/m
Equation (10) can be interpreted by saying that if the “position” and
“momentum” vectors of a wave packet are regarded as the average or
expectation values of these quantities, then the classical and quantum
motions will agree.
(b)
d < P x >
dt
=
d
dt
ψ
∗
−i
∂
∂ x
ψdτ = −i
d
dt
ψ
∗ ∂ψ
∂ x
dτ
= −i
dψ
∗
dt
∂ψ
∂ x
dτ +
ψ
∗ ∂
∂ x
∂ψ
∂t
dτ
(11)
Now i
∂ψ
∂t
= −
2
2m
∇
2
ψ + V ψ
−
i∂ψ
∗
∂t
= −
2
2m
∇
2
ψ
∗
+ V ψ
∗
(12)
Using (12) in (11)
d
dt
< P x >= −
ψ
∗ ∂
∂ x
−
2
2m
∇
2
ψ + V ψ
dτ
+
−
2
2m
∇
2
ψ + V ψ
∂ψ
∂ x
d τ