246
3 Quantum Mechanics – II
3.122 f (θ ) = −
μ
2π 2
V (r )e
iqr d
3 r
= −
μ
2π 2
∞
0
V (r )r
2 dr
+1
−1
e
iqr cos θ d(cos θ )
2π
0
dϕ
= −
μ
2π 2
V (r )r
2
(e
iqr
− e
−iqr )
iqr
2π dr
= −
2μ
q 2
∞
0
V (r )r sin(qr )dr
3.123 A =
∞
0
sin(qr /)
(qr /)
V (r )4πr
2 dr
Substitute V (r ) ∼
e
−r/R
r
where R = /mc
A ∼
∞
0
e
−r/R sin(qr /)
q
dr
Put 1/R = a and q/ = 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
3 Quantum Mechanics – II
3.122 f (θ ) = −
μ
2π 2
V (r )e
iqr d
3 r
= −
μ
2π 2
∞
0
V (r )r
2 dr
+1
−1
e
iqr cos θ d(cos θ )
2π
0
dϕ
= −
μ
2π 2
V (r )r
2
(e
iqr
− e
−iqr )
iqr
2π dr
= −
2μ
q 2
∞
0
V (r )r sin(qr )dr
3.123 A =
∞
0
sin(qr /)
(qr /)
V (r )4πr
2 dr
Substitute V (r ) ∼
e
−r/R
r
where R = /mc
A ∼
∞
0
e
−r/R sin(qr /)
q
dr
Put 1/R = a and q/ = 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