At high energy, the particles behave classically and see only the aspect disk with areaπb^2 , but at
low energy, thes-wave scattering sees the entire surface area of the sphere,which is 4πb^2.
In terms of the Bessel functions, it is possible to write downan exact expression for the cross
section, valid for all ranges of the momentak,
σtot=
4 π
k^2∑∞ℓ=0(2ℓ+ 1)
(jℓ(kb))^2
|hℓ(kb)|^2(12.113)
Collecting the first two terms,ℓ≤1, we find,
σtot=
4 π
k^2(
k^2 b^2 +3
1 +k^2 b^2(−kbcoskb+ sinkb)^2)
(12.114)At very high momentumkb≫1, we may use the asymptotic expressions for the Bessel functions
to evaluate the cross section, so that
(jℓ(kb))^2
|hℓ(kb)|^2
∼sin^2(
kb−πℓ
2)
(12.115)Summing overℓleads to a divergent total cross section! This is now a UV effect, operating at high
momentum scattering, and it is due to the fact that we are assuming that the sphere is perfectly
reflecting even to the highest momenta. Physically, this assumption is not valid in fact. Sakurai
proposes then to limit the sum overℓtoℓℓmax< kb, and refers to this as areasonable assumption.
Please judge for yourself.
12.13The hard spherical shell
Another very interesting special case is for aδ-function shell of finite strength, given by
U(r) =−U 0 δ(r−b) (12.116)whereU 0 may be either positive or negative. To solve for the phase shifts, we use the potential
U(r) above in the integral equation (12.78). The integral overr′is readily performed since the
δ(r′−b) potential localizes the integral atr′=b, and we find,
Rℓ(r) =jℓ(kr)−U 0 b^2 Gℓ(r,b)Rℓ(b) (12.117)To determineRℓ(b), we simply evaluate this equation atr=b, so that
Rℓ(b) =
jℓ(kb)
1 +iU 0 kb^2 jℓ(kb)hℓ(kb)(12.118)
This result gives an exact expression for the partial wave functionsRℓ(r) outside the spherical
shell, wherer > b(the inside region may be solved for as well but is immaterialfor the scattering
problem), and we find,
Rℓ(r) =jℓ(kr) +
iU 0 kb^2 jℓ(kb)^2 hℓ(kr)
1 −iU 0 kb^2 jℓ(kb)hℓ(kb)