1000 Solved Problems in Modern Physics

246 3 Quantum Mechanics – II

3.122 f(θ)=−

( μ

2 π^2

)∫

V(r)eiqrd^3 r

=−

( μ

2 π^2

)∫∞

0

V(r)r^2 dr

∫+ 1

− 1

eiqrcosθd(cosθ)

∫ 2 π

0

dφ

=−

( μ

2 π^2

)∫

V(r)r^2

[

(eiqr−e−iqr)

iqr

]

2 πdr

=−

(

2 μ

q^2

)∫∞

0

V(r)rsin(qr)dr

3.123 A=

∫∞

0

sin(qr/)

(qr/) V(r)4πr

(^2) dr
SubstituteV(r)∼e
−r/R
r
whereR=/mc

A∼

∫∞

0

e−r/R

sin(qr/)

q

dr

Put 1/R=aandq/=b

A∼

∫∞

0

e−ar

sin(br)

b

dr

=

1

b

[

b

a^2 +b^2

]

=

1

a^2 +b^2

=

1

1

R^2 +

q^2

^2

∼

1

^2

R^2 +q

2 =

1

q^2 +m^2 c^2