130_notes.dvi

The lowestℓNeumann functions(irregular at the origin) solutions are listed below.

n 0 (ρ) =−

cosρ

ρ

n 1 (ρ) =−

cosρ

ρ^2

−

sinρ

ρ

n 2 (ρ) =−

3 cosρ

ρ^3

−

3 sinρ

ρ^2

+

cosρ

ρ

The mostgeneral solution is a linear combinationof the Bessel and Neumann functions.
The Neumann function should not be used in a region containing the origin. The Bessel and
Neumann functions are analogous the sine and cosine functions of the 1D free particle solutions.
The linear combinations analogous to the complex exponentials of the1D free particle solutions are
thespherical Hankel functions.

h

(1)

ℓ (ρ) =jℓ(ρ) +inℓ(ρ) = (−ρ)

ℓ

(

1

ρ

d

dρ

)ℓ

sinρ−icosρ

ρ

→−

i

ρ

ei(ρ−

ℓπ 2 )

h(2)ℓ (ρ) =jℓ(ρ)−inℓ(ρ) =h(1)ℓ ∗(ρ)

The functional for for largeris given. TheHankel functions of the first typeare the ones that
will decay exponentially asrgoes to infinity ifE < V, so it isright for bound state solutions.

The lowestℓHankel functions of the first type are shown below.

h(1) 0 (ρ) =

eiρ

iρ

h(1) 1 (ρ) =−

eiρ

ρ

(

1 +

i

ρ

)

h(1) 2 (ρ) =

ieiρ

ρ

(

1 +

3 i

ρ

−

3

ρ^2

)

We should also give thelimits for larger, (ρ >> ℓ),of the Bessel and Neumann functions.

jℓ(ρ)→

sin

(

ρ−ℓπ 2

)

ρ

nℓ(ρ)→

cos

(

ρ−ℓπ 2

)

ρ

Decomposing the sine in the Bessel function at larger, we see that the Bessel function is composed
of an incoming spherical wave and an outgoing spherical wave of thesame magnitude.

jℓ(ρ)→−

1

2 ikr

(

e−i(kr−ℓπ/2)−ei(kr−ℓπ/2)

)

This is important. If the fluxes were not equal, probability would build up at the origin. All our
solutions must have equal flux in and out.