Spherical Bessel Functions *

We will now give the full solutions in terms of

\begin{displaymath}\bgroup\color{black}\rho=kr .\egroup\end{displaymath}

These are written for \bgroup\color{black}$E>V$\egroup but can be are also valid for \bgroup\color{black}$E<V$\egroup where \bgroup\color{black}$k$\egroup becomes imaginary.

\begin{displaymath}\bgroup\color{black}\rho=kr\rightarrow i\kappa r \egroup\end{displaymath}

The full regular solution of the radial equation for a constant potential for a given \bgroup\color{black}$\ell$\egroup is

\begin{displaymath}\bgroup\color{black}j_\ell(\rho)=(-\rho)^\ell\left({1\over\rho}{d\over d\rho}\right)^\ell{\sin\rho\over\rho} \egroup\end{displaymath}

the spherical Bessel function. For small \bgroup\color{black}$r$\egroup, the Bessel function has the following behavior.

\begin{displaymath}\bgroup\color{black} j_\ell(\rho)\rightarrow {\rho^\ell\over 1\cdot 3\cdot 5\cdot ... (2\ell+1)} \egroup\end{displaymath}

The full irregular solution of the radial equation for a constant potential for a given \bgroup\color{black}$\ell$\egroup is

\begin{displaymath}\bgroup\color{black}n_\ell(\rho)=-(-\rho)^\ell\left({1\over\rho}{d\over d\rho}\right)^\ell{\cos\rho\over\rho} \egroup\end{displaymath}

the spherical Neumann function. For small \bgroup\color{black}$r$\egroup, the Neumann function has the following behavior.

\begin{displaymath}\bgroup\color{black} n_\ell(\rho)\rightarrow {1\cdot 3\cdot 5\cdot ... (2\ell+1)\over \rho^{\ell+1}} \egroup\end{displaymath}

The lowest \bgroup\color{black}$\ell$\egroup Bessel functions (regular at the origin) solutions are listed below.

\bgroup\color{black}$\displaystyle j_0(\rho)={\sin\rho\over\rho} $\egroup
\bgroup\color{black}$\displaystyle j_1(\rho)={\sin\rho\over\rho^2}-{\cos\rho\over\rho} $\egroup
\bgroup\color{black}$\displaystyle j_2(\rho)={3\sin\rho\over\rho^3}-{3\cos\rho\over\rho^2}-{\sin\rho\over\rho} $\egroup
The lowest \bgroup\color{black}$\ell$\egroup Neumann functions (irregular at the origin) solutions are listed below.
\bgroup\color{black}$\displaystyle n_0(\rho)=-{\cos\rho\over\rho} $\egroup
\bgroup\color{black}$\displaystyle n_1(\rho)=-{\cos\rho\over\rho^2}-{\sin\rho\over\rho} $\egroup
\bgroup\color{black}$\displaystyle n_2(\rho)=-{3\cos\rho\over\rho^3}-{3\sin\rho\over\rho^2}+{\cos\rho\over\rho} $\egroup

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.

\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}


\begin{displaymath}\bgroup\color{black}h^{(2)}_\ell(\rho)=j_\ell(\rho)-i n_\ell(\rho)=h^{(1)*}_\ell(\rho) \egroup\end{displaymath}

The functional for for large \bgroup\color{black}$r$\egroup is given. The Hankel functions of the first type are the ones that will decay exponentially as \bgroup\color{black}$r$\egroup goes to infinity if \bgroup\color{black}$E<V$\egroup, so it is right for bound state solutions.

The lowest \bgroup\color{black}$\ell$\egroup Hankel functions of the first type are shown below.

\bgroup\color{black}$\displaystyle h^{(1)}_0(\rho)={e^{i\rho}\over i\rho} $\egroup
\bgroup\color{black}$\displaystyle h^{(1)}_1(\rho)=-{e^{i\rho}\over \rho}\left(1+{i\over\rho}\right) $\egroup
\bgroup\color{black}$\displaystyle h^{(1)}_2(\rho)={ie^{i\rho}\over \rho}\left(1+{3i\over\rho}-{3\over\rho^2}\right) $\egroup

We should also give the limits for large $r$, \bgroup\color{black}$(\rho»\ell)$\egroup,of the Bessel and Neumann functions.

\begin{eqnarray*}
j_\ell(\rho)\rightarrow {\sin\left(\rho-{\ell\pi\over 2}\right...
...o)\rightarrow {\cos\left(\rho-{\ell\pi\over 2}\right)\over\rho}
\end{eqnarray*}


Decomposing the sine in the Bessel function at large \bgroup\color{black}$r$\egroup, we see that the Bessel function is composed of an incoming spherical wave and an outgoing spherical wave of the same magnitude.

\begin{displaymath}\bgroup\color{black}j_\ell(\rho)\rightarrow -{1\over 2ikr}\left(e^{-i(kr-\ell\pi/2)}-e^{i(kr-\ell\pi/2)}\right) \egroup\end{displaymath}

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