QuantumPhysics.dvi

For real values ofρ, the functionjℓ(ρ) is real, buthℓ(ρ) is complex. The asymptotic values of these
functions are as follows. Asρ→0, we have,

jℓ(ρ) ∼

ρℓ

(2ℓ+ 1)!!

hℓ(ρ) ∼ −i

(2ℓ−1)!!

ρℓ+1

(12.71)

while forρ→∞, we have,

jℓ(ρ) ∼

1

ρ

cos

(

ρ−

π

2

(ℓ+ 1)

)

hℓ(ρ) ∼ −

i

ρ

eiρ−iπℓ/^2 (12.72)

Note that the functionhℓ(ρ) is sometimes denoted ash(1)ℓ (ρ), whileh(2)(ρ) = (h(1)(ρ∗))∗.

12.10.2Partial wave expansion of wave functions

The free wave solutionφ(x) admits an expansion in terms ofjℓ, given as follows,

eik·r=eikrcosθ=

∑∞

ℓ=0

(i)ℓ(2ℓ+ 1)jℓ(kr)Pℓ(cosθ) (12.73)

wherePℓ(cosθ) are the Legendre polynomials, familiar from the structureof spherical harmonics.
More generally now, we can express any wave function for the spherical potential problem in such
an expansion in terms of spherical harmonics,

φk(r) =

1

(2π)^3 /^2

∑∞

ℓ=0

(i)ℓ(2ℓ+ 1)jℓ(kr)Pℓ(cosθ)

ψk(r) =

1

(2π)^3 /^2

∑∞

ℓ=0

(i)ℓ(2ℓ+ 1)Rℓ(kr)Pℓ(cosθ) (12.74)

Note that the expansion involves only the spherical harmonicsYℓ^0. The reason is that while the
Schr ̈odinger equation is invariant under all rotations, the initial conditions (namely given by the
incoming plane wave) break this symmetry to rotations around the incoming wave vectorkonly.
Thus, the wave functions can have no dependence onφby symmetry, which prevents the occurrence
ofYℓmform 6 = 0.

Instead of substituting these expressions into the integral equation we had derived earlier, it
turns out to be much more convenient to work from the initial Schr ̈odinger equation, which we
express as follows,

(

d^2

dr^2

+

2

r

d

dr

−

ℓ(ℓ+ 1)

r^2

+k^2

)

Rℓ(r) =U(r)Rℓ(r) (12.75)