1000 Solved Problems in Modern Physics

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)

3

2

∫∞

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