QuantumPhysics.dvi

The first term represents a radially incoming wave, while thesecond represents a radially outgoing
wave. Because the potentialV(r) is spherically symmetric, the probability in each of the partial
waves separately is conserved. Thus, the probability currents of the incoming and outgoing radial
waves have to be equal. This requires that the moduli of the incoming and outgoing waves coincide.
Therefore, the composite factor of the second term must in fact be a phase,

1 − 2 ik

∫∞

0

dr′(r′)^2 jℓ(kr′)U(r′)Rℓ(r′) =e^2 iδℓ(k) (12.91)

forδℓ(k) a real function ofk, for eachℓ. The asymptotic form of the wave function thus becomes,

Rℓ(r) ∼

1

2 ikr

{

−e−ikr+iπℓ/^2 +eikr−iπℓ/2+2iδℓ(k)

}

∼

eiδℓ(k)

kr

sin

(

kr−

πℓ

2

+δℓ(k)

)

(12.92)

The anglesδℓ(k) are referred to as thephase shift for angular momentumℓ.

Alternatively, we may expose inRℓ(r) the contribution from the incoming plane wave, by
isolating the functionjℓ(kr). As a result, we have, asr→∞,

Rℓ(r)∼jℓ(kr) +

eikr−iπℓ/^2

r

×

e^2 iδℓ(k)− 1

2 ik

(12.93)

Therefore, we may now derive a partial wave expansion for thescattering amplitudef(k′,k), by
comparing the general definition off(k′,k) with the partial wave expansion obtained here. The
definition is given by the behavior of the wave functionψk(r) for larger,

ψk(r)∼

1

(2π)^3 /^2

{

eik·r+

eikr

r

f(k′,k)

}

(12.94)

The partial wave result in the same limit obtained here is given by,

ψk(r)−φk(r) =

eikr

(2π)^3 /^22 ikr

∑∞

ℓ=0

(2ℓ+ 1)

(

e^2 iδℓ(k)− 1

)

Pℓ(cosθ) (12.95)

As a result, we obtain the partial wave expansion off(k′,k),

f(k′,k) =

1

2 ik

∑∞

ℓ=0

(2ℓ+ 1)

(

e^2 iδℓ(k)− 1

)

Pℓ(cosθ) (12.96)

The total cross section may be computed by performing the angular integrations of|f(k′,k)|^2. To
do so, we use the orthogonality of the Legendre polynomials as well as their normalizations,

∫

4 π

dΩPℓ(cosθ)Pℓ′(cosθ) =

4 πδℓ,ℓ′

2 ℓ+ 1

(12.97)