Solutions to the Radial Equation for Constant Potentials

Solutions to the radial equation in a constant potential are important since they are the solutions for large \bgroup\color{black}$r$\egroup in potentials of limitted range. They are therefore used in scattering problems as the incoming and outgoing states. The solutions are the spherical Bessel and spherical Neumann functions.

\rightarrow{\sin(\rho-{\ell\pi\over 2})\over\rho}\egroup\end{displaymath}

\rightarrow{-\cos(\rho-{\ell\pi\over 2})\over\rho}\egroup\end{displaymath}

where \bgroup\color{black}$\rho=kr$\egroup. The linear combination of these which falls off properly at large \bgroup\color{black}$r$\egroup is called the Hankel function of the first type.

\begin{displaymath}\bgroup\color{black}h^{(1)}_\ell(\rho)=j_\ell(\rho)+i n_\ell(...
\rightarrow -{i\over\rho}e^{i(\rho-{\ell\pi\over 2})}\egroup\end{displaymath}

We use these solutions to do a partial wave analysis of scattering, solve for bound states of a spherical potential well, solve for bound states of an infinite spherical well (a spherical ``box''), and solve for scattering from a spherical potential well.

Jim Branson 2013-04-22