Due May 14
1) In class and in Griffiths the problem of hard-sphere scattering was considered. It was
found that the partial wave coefficients )(
)(
)12(4 )1( kah
kaj
lic
l
l
l
l+−=
π
where a is the
radius of the sphere.
a) Using the explicit form for )(
0kaj and )(
)1(
0kah show that aki
eakic −
=)sin(4
0
π
.
b) Show that this means that the phase-shift is given by ak
−=
0
δ
. The fact that we
can define a (real) phase shift implies means the value of 0
chas an absolute upper
bound. This bound followed from unitarity (conservation of particle number).
Thus our solution is consistent with unitarity.
c) At very low incident momentum (k a <<1) the l=0 partial wave dominates the
scattering. Show that as k a goes to zero the differential cross-section is given by
2
a
d
d=
Ω
σ
, independent of angle.
2) Use the result of part 1.c to show that if a total of g particles per unit area are incident
on the target (where )(tfluxtdg ∫
=) with low incident momentum, then the total
number of particles scattered in the backward hemisphere (θ > π/2) is given by
2
2ag
π
.
Problem 11.4