QuantumPhysics.dvi

the above relation becomes,

e^2 iδℓ(k)=−

h∗ℓ(kb)

hℓ(kb)

(12.106)

The phase shifts for the lowest values ofℓmay be computed explicitly, using the expressions for
the corresponding Hankel functions,

h 0 (ρ) = −i

eiρ

ρ

h 1 (ρ) = −i

eiρ

ρ^2

(1−iρ)

h 2 (ρ) = −i

eiρ

ρ^3

(

−3 + 3iρ+ρ^2

)

(12.107)

As a result, we have

e^2 iδ^0 (k) = e−^2 ikb

e^2 iδ^1 (k) = e−^2 ikb×

1 +ikb

1 −ikb

e^2 iδ^2 (k) = e−^2 ikb×

− 3 − 3 ikb+k^2 b^2

−3 + 3ikb+k^2 b^2

(12.108)

or taking the logs,

δ 0 (k) = −kb

δ 1 (k) = −kb+ Arctg(kb)

δ 2 (k) = −kb+ Arctg

( kb

1 −k^2 b^2 / 3

)

(12.109)

Note that in the low momentum limit,kb→0, one may obtain an approximate result valid for all
ℓ, using the asymptotic expansions of the spherical Bessel functions,

jℓ(kb) = Re(hℓ(kb))∼

(kb)ℓ

(2ℓ+ 1)!!

nℓ(kb) = Im(hℓ(kb))∼−

(2ℓ−1)!!

(kb)ℓ+1

(12.110)

Sincejℓ(kb)≪nℓ(kb) in this limit, we find that

δℓ(k)∼

jℓ(kb)

nℓ(kb)

∼−

(kb)^2 ℓ+1

(2ℓ+ 1)!!(2ℓ−1)!!

(12.111)

Thus, forkb≪1, it makes sense to neglectℓ≥1 in the low momentum limit, and we find

σtot∼

4 π

k^2

sin^2 δ 0 ∼ 4 πb^2 (12.112)