QuantumPhysics.dvi

Differentiating a second time, and substituting the result into the defining equation (12.77) for
Gℓ(r,r′), we obtain,
(
d^2
dr^2

+

2

r

d

dr

−

ℓ(ℓ+ 1)

r^2

+k^2

)

Gℓ(r,r′) = 2kCℓδ(r−r′)W(kr) (12.83)

where

W(x) =jℓ(x)h′ℓ(x)−j′ℓ(x)hℓ(x) (12.84)

This quantity is the Wronskian for the Bessel equation. The key property of the Wronskian is that
it satisfies a first order linear differential equation which may always be integrated. This differential
equation may be deduced directly from the second order equation, namely,

W′(x) =jℓ(x)h′′ℓ(x)−j′′ℓ(x)hℓ(x) =−

2

x

W(x) (12.85)

and solved by

W(x) =

W 0

x^2

(12.86)

whereW 0 is a constant. Actually, even this constant may be calculated, using the asymptotic
behavior ofjℓand ofhℓ, and we findW 0 =i/2. Putting all together, we find that
(
d^2
dr^2

+

2

r

d

dr

−

ℓ(ℓ+ 1)

r^2

+k^2

)

Gℓ(r,r′) =kCℓδ(r−r′)

i

k^2 r^2

(12.87)

Hence, the desired normalization forGℓis recovered by settingCℓ=−ik, and we have,

Gℓ(r,r′) =−ik

(

θ(r′−r)jℓ(kr)hℓ(kr′) +θ(r−r′)hℓ(kr)jℓ(kr′)

)

(12.88)

12.11Phase shifts

For asymptotically large distancer, and a potential that vanishes at infinityV(r)→0 asr→∞,
it is only the second term in the Green functionGℓ(r,r′) of (12.88) that will contribute. This may
be seen by decomposing the integration region overr′into two pieces,

Rℓ(r) = jℓ(kr)−ikhℓ(kr)

∫r

0

dr′(r′)^2 jℓ(kr′)U(r′)Rℓ(r′)

−ikjℓ(kr)

∫∞

r

dr′(r′)^2 hℓ(kr′)U(r′)Rℓ(r′) (12.89)

Sincehℓ(kr′) andRℓ(kr′) fall off as 1/r′asr′→∞, we see thatU(r′) should go to zero faster than
1 /r′. Under this assumption, the second term may be neglected asr→ ∞, and we may use the
asymptotic behaviors of the Bessel functions to obtain, asr→∞,

Rℓ(r) ∼

1

kr

sin

(

kr−

ℓπ

2

)

−

eikr−iπℓ/^2

r

∫∞

0

dr′(r′)^2 jℓ(kr′)U(r′)Rℓ(r′) (12.90)

∼

1

2 ikr

{

−e−ikr+iπℓ/^2 +eikr−iπℓ/^2

(

1 − 2 ik

∫∞

0

dr′(r′)^2 jℓ(kr′)U(r′)Rℓ(r′)

)}