3.3 Solutions 157
3.3 Normalization condition is
∫∞−∞|ψ|^2 dx= 1N^2
∫∞
−∞(x^2 +a^2 )−^2 dx= 1Putx=atanθ;dx=sec^2 θdθ
(
2 N^2
α^3)∫π/ 20cos^2 θdθ=N^2 π/ 2 a^3 = 1ThereforeN=(
2 a^3
π) 1 / 2
3.4ψ=Aeikx+Be−ikx
The fluxJx=(
2 im)[
ψ∗ddψx−(
dψ∗
dx)
ψ]=(
2 im)
[(
Ae−ikx+Beikx)
ik(
Aeikx−Be−ikx)+ik(
Ae−ikx−Beikx)(
Aeikx+Be−ikx)]=(
k
2 m)
[
A^2 −B^2 −ABe−^2 ikx+ABe^2 ikx+A^2 −B^2 +ABe−^2 ikx−ABe^2 ikx]=(
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 byφ∗and (2) byφand 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/ 2 im
1
2 im∇·
(
φ∗∇φ−φ∇φ∗)
−
1
2 im∂
∂t(
φ∗∂φ
∂t−φ∂φ∗
∂t)
= 0
∇·
[
1
2 im(
φ∗∇φ−φ∇φ∗)
]
+
∂
∂t[
i
2 m(
φ∗∂φ
∂t−φ∂φ∗
∂t)]
= 0
Or∇·J+∂ρ
∂t