Computing the Radial Wavefunctions *

The radial wavefunctions are given by

\begin{displaymath}\bgroup\color{black}R(\rho)=\rho^\ell\sum\limits_{k=0}^{n-\ell-1} a_k\rho^k e^{-\rho/2} \egroup\end{displaymath}

where

\begin{displaymath}\bgroup\color{black} \rho={2Z\over na_0}r \egroup\end{displaymath}

and the coefficients come from the recursion relation

\begin{displaymath}\bgroup\color{black}a_{k+1}={k+\ell+1-n\over (k+1)(k+2\ell+2)}a_k .\egroup\end{displaymath}

The series terminates for \bgroup\color{black}$k=n-\ell-1$\egroup.

Lets start with \bgroup\color{black}$R_{10}$\egroup.

\begin{eqnarray*}
R_{10}(r)=\rho^0\sum\limits_{k=0}^{0} a_k\rho^k e^{-\rho/2} \\
R_{10}(r)= C e^{-Zr/a_0} \\
\end{eqnarray*}


We determine \bgroup\color{black}$C$\egroup from the normalization condition.

\begin{eqnarray*}
\int\limits_0^\infty r^2 R^*_{n\ell}R_{n\ell}\; dr=1 \\
\vert C\vert^2\int\limits_0^\infty r^2 e^{-2Zr/a_0}\; dr=1 \\
\end{eqnarray*}


This can be integrated by parts twice.

\begin{eqnarray*}
2\left({a_0\over 2Z}\right)^2\vert C\vert^2\int\limits_0^\inft...
...
R_{10}(r)=2\left({Z\over a_0}\right)^{3\over 2} e^{-Zr/a_0} \\
\end{eqnarray*}


\bgroup\color{black}$R_{21}$\egroup can be computed in a similar way. No recursion is needed.

Lets try \bgroup\color{black}$R_{20}$\egroup.

\begin{eqnarray*}
R_{20}(r)=\rho^0\sum\limits_{k=0}^{1} a_k\rho^k e^{-\rho/2} \\...
...0 \\
R_{20}(r)= C\left(1-{Zr\over 2a_0}\right) e^{-Zr/2a_0} \\
\end{eqnarray*}


We again normalize to determine the constant.



Jim Branson 2013-04-22