PHYS 402 Homework---Due May 6
1. Consider scattering of a potential of the form 2
0
)( r
eVrV
λ
−
=
r
.
a. Use the Born approximation to show that at sufficiently low energies and
sufficiently small 0
V, the scattering is isostropic . State the condition which
specifies sufficiently low energies and sufficiently small 0
V for the Born
approximation to be valid and for the potential to be isotropic.
b. Compute the differential cross-section Ω
d
d
σ
in the Born approximation
c. Show that only phase shift contributing is the l=0 and find the phase shifts by
direct comparison with the scattering amplitude. (You do not need to solve the
Schrodinger equation to do this.
d. In computing the result in part b. you computed a scattering amplitude in the Born
approximation which, if done correctly, was entirely real. The optical theorem
says that the total cross-section is given by
(
)
k
f0(Im4
=
=
θ
π
σ
. Clearly the
total cross section in a. is nonzero. How do you reconcile these two facts.
2. Show that in the regime where the Born Approximation is valid, the differential cross-
section for back-scattering (θ = π) at an incident energy 0
E is the same as differential
cross-section for scattering in the perpendicular direction (θ = π/2) at an incident energy
0
2E. (Hint: think about momentum transfer).
3. In class we showed that in the Born approximation scattering from a Yukowa potential of
the form
r
e
CrV ar /
)(
λ
−
=
r, yields a scattering amplitude of
( )
2
1
2
2
2
a
q
Cm
f+
−= h
where q is the magnitude of the momentum transfer. We have also argued that at high
energies the scattering should be forward peaked. The purpose of this problem is to
demonstrate this explicitly.
a. Show that the maximum of the differential cross-section is maximal in the
forward direction (θ = 0) and is is given by 4
422
max
4
h
aCm
d
d=
Ω
σ
.
b. Show that the angle at which the differential cross-section is reduced from the
maximum by a factor of four is given by
(
)
akak 1
21
1
sin2 ≈= −
θ
where the last
equality holds for 1
>>
ak . Explain why this result implies extreme forward
peaking at very high energies.
