3.3 Solutions
165
Integrating by parts twice
d
dt
< P x >= −
ψ
∗
∂
∂ x
(V ψ) − V
∂ψ
∂ x
dτ
= −
ψ
∗ ∂ V
∂ x
ψdτ =<
−∂ V
∂ x
>
These two examples support the correspondence principle as they show
that the wave packet moves like a classical particle provided the expectation value gives a good representation of the classical variable.
3.15 (a) Using the Laplacian in the time-independent Schrodinger equation
−
2
2m
1
r 2
∂
∂r
r
2 ∂
∂r
+
1
r 2 sin θ
∂
∂θ
sin θ
∂
∂θ
+
1
r 2 sin
2
θ
∂
2
∂ϕ 2
ψ(r, θ, ϕ) + V (r )ψ(r, θ, ϕ) = Eψ(r, θ, ϕ)
( 1 )
We solve this equation by method of separation of variables
Let ψ (r, θ, ϕ) = ψ r (r ) Y (θ, ϕ)
( 2 )
Use (2) in (1) and multiply by
−
2m
2 .r
2
/ψ r (r )Y (θ, ϕ) and rearrange
1
ψ r
(r )
d
dr
r
2 dψ r (r )/dr
+
2mr
2
2 [E − V (r )]
= −
1
Y
(θ, ϕ)
1
sin θ
∂
∂θ
sin θ
∂
∂θ
(sin θ∂Y (θ, ϕ) /∂θ) +
1
sin
2
θ
∂
2 Y (θ, ϕ) /∂ϕ
2
(3)
It is assumed that V (r ) depends only on r .
L.H.S. is a function of r only and R.H.S is a function of θ and ϕ only.
Then each side must be equal to a constant, say λ.
1
sin θ
∂
∂θ
sin θ
∂Y
∂θ
θ, ϕ
+
1
sin
2
θ
∂
2 Y
∂ϕ 2
θ, ϕ
+ λY (θ, ϕ) = 0
( 4 )
The radial equation is
d
dr
r
2 dψ r (r )
dr
+
2mr
2
2 [E − V (r ) − λ] ψ r (r ) = 0
( 5 )
(b) The angular equation (4) can be further separated by substituting
Y (θ, ϕ) = f (θ )g(θ )
( 6 )