130_notes.dvi

Higher powers ofrare OK, but are not dominant. Plugging this into the equation we get
[
s(s−1)rs−^2 + 2srs−^2

]

−ℓ(ℓ+ 1)rs−^2 = 0.

s(s−1) + 2s=ℓ(ℓ+ 1)

s(s+ 1) =ℓ(ℓ+ 1)

There are actually two solutions to this equation,s=ℓands=−ℓ−1. The first solution,s=ℓ, is
well behaved at the origin (regular solution). The second solution,s=−ℓ−1, causes normalization
problems at the origin (irregular solution).

15.3 Spherical Bessel Functions*.

We will now give the full solutions in terms of

ρ=kr.

These are written forE > Vbut can be are also valid forE < V wherekbecomes imaginary.

ρ=kr→iκr

Thefull regular solutionof the radial equation for a constant potential for a givenℓis

jℓ(ρ) = (−ρ)ℓ

(

1

ρ

d

dρ

)ℓ

sinρ

ρ

thespherical Bessel function. For smallr, the Bessel function has the following behavior.

jℓ(ρ)→

ρℓ

1 · 3 · 5 ·...(2ℓ+ 1)

The fullirregular solutionof the radial equation for a constant potential for a givenℓis

nℓ(ρ) =−(−ρ)ℓ

(

1

ρ

d

dρ

)ℓ

cosρ

ρ

thespherical Neumann function. For smallr, the Neumann function has the following behavior.

nℓ(ρ)→

1 · 3 · 5 ·...(2ℓ+ 1)

ρℓ+1

The lowestℓBessel functions(regular at the origin) solutions are listed below.

j 0 (ρ) =

sinρ

ρ

j 1 (ρ) =

sinρ

ρ^2

−

cosρ

ρ

j 2 (ρ) =

3 sinρ

ρ^3

−

3 cosρ

ρ^2

−

sinρ

ρ