##

Spherical Bessel Functions *****

We will now give the full solutions in terms of

These are written for
but can be are also valid for
where
becomes imaginary.

The **full regular solution** of the radial equation for a constant potential for a given
is

the **spherical Bessel function**.
For small
, the Bessel function has the following behavior.

The full **irregular solution** of the radial equation for a constant potential for a given
is

the **spherical Neumann function**.
For small
, the Neumann function has the following behavior.

The lowest
**Bessel functions** (regular at the origin) solutions are listed below.

The lowest
**Neumann functions** (irregular at the origin) solutions are listed below.
The most **general solution is a linear combination** of 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 the 1D free particle solutions are the
**spherical Hankel functions**.

The functional for for large
is given.
The **Hankel functions of the first type** are the ones that will decay exponentially as
goes to
infinity if
, so it is **right for bound state solutions**.
The lowest
Hankel functions of the first type are shown below.

We should also give the **limits for large **,
,of the Bessel and Neumann functions.

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

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.

Jim Branson
2013-04-22