The full regular solution of the radial equation for a constant potential for a given
The full irregular solution of the radial equation for a constant potential for a given
The lowest Bessel functions (regular 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 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.
Jim Branson 2013-04-22