3.3 Solutions
157
3.3 Normalization condition is
∞
−∞
|ψ|
2 dx = 1
N
2
∞
−∞
(x
2
+ a
2 )
−2 dx = 1
Put x = a tan θ; dx = sec
2
θ dθ
2N
2
α 3
π/2
0
cos
2
θ d θ = N
2
π/2a
3
= 1
Therefore N =
2a
3
π
1/2
3.4 ψ = Ae
ikx
+ Be
−ikx
The flux J x =
2im
ψ
∗ dψ
dx
−
dψ ∗
dx
ψ
=
2im
Ae
−ikx
+ Be
ikx
ik
Ae
ikx
− Be
−ikx
+ ik
Ae
−ikx
− Be
ikx
Ae
ikx
+ Be
−ikx
=
k
2m
A
2
− B
2
− ABe
−2ikx
+ ABe
2ikx
+ A
2
− B
2
+ ABe
−2ikx
− ABe
2ikx
=
k
m
A
2
− B
2
3.5 In natural units ( = c = 1) Klein – Gordon equation is
∇
2
ϕ −
∂
2
ϕ
dt 2 − m
2
ϕ = 0
( 1 )
The complex conjugate equation is
∇
2
ϕ
∗
−
∂
2
ϕ
∗
∂t 2 − m
2
ϕ
∗
= 0
( 2 )
Multiplying (1) from left by ϕ
∗ and (2) by ϕ and subtracting (1) from (2)
ϕ∇
2
ϕ
∗
− ϕ
∗
∇
2
ϕ − ϕ
∂
2
ϕ
∗
∂t 2 − ϕ
∂
2
ϕ
∗
∂t 2 + ϕ
∗ ∂
2
ϕ
∂t 2 = 0
∇.
ϕ∇ϕ
∗
∇ϕ
−
∂
∂t
ϕ
∂ϕ
∗
∂t
− ϕ
∗ ∂ϕ
∂t
= 0
Changing the sign through out and multiplying by 1/2im
1
2im
∇ ·
ϕ
∗
∇ϕ − ϕ∇ϕ
∗
−
1
2im
∂
∂t
ϕ ∗
∂ϕ
∂t
− ϕ
∂ϕ
∗
∂t
= 0
∇ ·
1
2im
ϕ
∗
∇ϕ − ϕ∇ϕ
∗
+
∂
∂t
i
2m
ϕ
∗ ∂ϕ
∂t
− ϕ
∂ϕ
∗
∂t
= 0
Or ∇ · J +
∂ρ
∂t
= 0