3.3 Solutions 213
Fig. 3.25Probability
distribution of electron in
n=2 orbit of H-atom
hν= 0 .088 MeV
λ= 8.^12418 × 104 nm= 0 .0141 nm
(b) The transition probability per unit time is
A∝ω^3 |rkk′|^2
For hydrogen-like atoms, such as the meisc atom
|rkk′|∝
1
z
,ω∝mμZ^2 ,so that
A∝m^3 μZ^4
The mean life time of the mesic atom in the 3dstate is
τμ=
(
AH
Aμ
)
τH=
(
me
mμ
) 3
τH/Z^4
=
(
0. 511
106
) 3
(
1. 6 × 10 −^8
)
(
1
154
)
= 3. 5 × 10 −^20 s.
3.74 Takekalong thez-axis so thatp.r/=k.r=krcosθ. Write
dτ=r^2 drd(cosθ)dφ
ψ(p)=
1
(2π)
(^32)
∫∞
r= 0
∫π
θ= 0
∫ 2 π
φ= 0
e−ikrcosθ
(
πa 03
)−^12
exp
(
−
r
a 0
)
r^2 drd(cosθ)dφ
(1)
∫ 2 π
0
dφ= 2 π (2)
∫+ 1
− 1
e−ikrcosθd(cosθ)=
(
2
kr
)
sinkr (3)
With the aid of (2) and (3), (1) becomes
ψ(p)=
(√
2
πk
)
(a 0 )−
3
2
∫∞
0
rsinkrexp
(
−
r
a 0
)
dr (4)
Now
∫∞
0 rsinkr e
−br=−∂
∂b
∫∞
0 sinkr e
−br