156
3 Quantum Mechanics – II
3.123 Using the Born approximation, the amplitude of scattering by a spherically
symmetric potential V (r ) with a momentum transfer q is given by
A =
∞
0
sin
qr
qr
V (r )4πr
2 dr
Show that in the case of a Yukawa-type potential, this leads to an amplitude
proportional to (q
2
+ m
2 c
2 )
−1 .
3.3 Solutions
3.3.1 Wave Function
3.1 E n =
n
2 h
2
8m L 2 =
π
2 n
2
2 c
2
2mc 2 L 2
=
π
2
× (197.3 MeV − fm)
2 n
2
2x0.511(MeV) × (10 6 fm) 2 = 0.038 n
2 eV
E 1 = 0.038 eV, E 2 = 0.152 eV, E 3 = 0.342 eV, E 4 = 0.608 eV
Δ E 43 = E 4 − E 3 = 0.608 − 0.342 = 0.266 eV
λ =
1, 241
0.266
= 4,665 nm
3.2 ψ(x) = (π/α)
−1/4 exp
−
α
2
2
x
2
Var x =< x
2
> − < x >
2
The expectation value
< x >=
∞
−∞
ψ
∗ x ψ dx = 0
because ψ and also ψ
∗ are even functions while x is an odd function. Therefore the integrand is an odd function
< x
2
>=
π
α
−1/2 ∞
−∞
x
2 exp(−α
2 x
2 )dx
Put α
2 x
2
= y; dx =
1
/ 2 α
√
y
< x
2
>=
πα
5
−1/2
∞
0
y
1/2 e
−y dy
But
∞
0 y
1/2 e
−y dy = Γ(3/2) =
√ π/2
Var x =< x
2
>= (4 α
5 )
−1/2