3.3 Solutions
213
Fig. 3.25 Probability
distribution of electron in
n = 2 orbit of H-atom
hν = 0.088 MeV
λ =
1241
8.8×10 4 nm = 0.0141 nm
(b) The transition probability per unit time is
A ∝ ω
3
|r kk |
2
For hydrogen-like atoms, such as the meisc atom
|r kk | ∝
1
z
, ω ∝ m μ Z
2
, so that
A ∝ m
3
μ Z
4
The mean life time of the mesic atom in the 3d state is
τ μ =
A H
A μ
τ H =
m e
m μ
3
τ H /Z
4
=
0.511
106
3
1.6 × 10
−8
1
15 4
= 3.5 × 10
−20 s.
3.74 Take k along the z-axis so that p.r/ = k.r = kr cos θ. Write
dτ = r
2 dr d(cos θ)dϕ
ψ( p) =
1
(2π)
3
2
∞
r=0
π
θ=0
2π
ϕ=0
e
−ikr cos θ
πa
3
0
−
1
2 exp
−
r
a 0
r
2 dr d (cos θ ) dϕ
(1)
2π
0
dϕ = 2π
(2)
+1
−1
e
−ikr cos θ d (cos θ) =
2
kr
sin kr
(3)
With the aid of (2) and (3), (1) becomes
ψ( p) =
√
2
π k
(a 0 )
−
3
2
∞
0
r sin kr exp
−
r
a 0
dr
(4)
Now
∞
0 r sin kr e
−br
=
−∂
∂b
∞
0 sin kr e
−br