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\)
Section 16.4 General Solution of the Radial Equation
The general solution of
(16.2.12) is the associated Laguerre polynomial. First let’s look at the
ordinary Laguerre polynomials of degree
\(q\text{,}\) which are obtained from Rodrigues’ formula
\begin{equation}
L_q(\rho) = e^\rho \frac{d^q}{d\rho^q}\left(\rho^q e^{-\rho}\right)\tag{16.4.1}
\end{equation}
The associated Laguerre polynomials are then defined as
\begin{equation}
L_q^p(\rho) = \frac{d^p}{d\rho^p} L_q(\rho)\tag{16.4.2}
\end{equation}
which is a polynomial of degree
\(q-p\text{.}\) The general solution of
(16.2.12) is the associated Laguerre polynomial:
\begin{equation}
H(\rho) = L_{n+\ell}^{2\ell+1}(\rho)\tag{16.4.3}
\end{equation}
a polynomial of degree
\((n+\ell)-(2\ell+1)=n-\ell-1\text{,}\) as expected corresponding to the value of
\(j_\text{max}\) given by
(16.3.11) .
The complete solution for \(R(r)\) is then
\begin{equation}
R_{n\ell}(r)
= N_{n\ell}\, r^\ell\, e^{-Zr/na_0}\, L_{n+\ell}^{2\ell+1}(2Zr/na_0)\tag{16.4.4}
\end{equation}
where \(N_{n\ell}\) is a normalization constant. Because the \(Y_\ell^m(\theta,\phi)\) are separately normalized, the normalization of \(\psi(r,\theta,\phi)=R_{n\ell}(r)Y_\ell^m(\theta,\phi)\) then requires that the \(R_{n\ell}(r)\) be normalized:
\begin{equation}
\int_0^\infty r^2 dr\, \vert R_{n\ell}(r)\vert^2 = 1\tag{16.4.5}
\end{equation}
Following a tedious but straightforward calculation,
\begin{equation}
N_{n\ell}
= \left[ \left( \frac{2Z}{na_0}\right)^3
\frac{(n-\ell-1)!}{2n[(n+\ell)!]^3} \right]^{1/2}
\left( \frac{2Z}{na_0} \right)^\ell\tag{16.4.6}
\end{equation}
so that the complete solution for \(R(r)\) is then
\begin{align}
R_{n\ell}(r)
\amp = \left[ \left( \frac{2Z}{na_0}\right)^3
\frac{(n-\ell-1)!}{2n[(n+\ell)!]^3} \right]^{1/2}\notag\\
\amp\qquad e^{-Zr/na_0} \left( \frac{2Zr}{na_0} \right)^\ell
L_{n+\ell}^{2\ell+1}(2Zr/na_0)\tag{16.4.7}
\end{align}
A table of the lowest energy radial wave functions can be found in
Section B.3 .
