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% $BNL;RJ*M}3X(B : $B1i=,LdBj(B 9-4 $B2rEz(B
% $B:n(B  $B@.(B  $B<T(B : g240897 $B5Z@n0l@?(B
% $B:n@.G/7nF|(B : 2002/10/31
% $BEE;R%a!<%k(B : g240897@mail.ecc.u-tokyo.ac.jp
% 
\mondai
{g240897 $B5Z@n0l@?(B}
{2002/10/31}
{g240897@mail.ecc.u-tokyo.ac.jp}
{9-4}
{
$B%i%2!<%k$NGfB?9`<0(B$L_1^1,L_2^1,\cdots$$B$N7A$r5a$a!"?eAG86;R$N(B$1s,2s,2p$$B50F;$NF07BJ}8~$N4X?t7A$r<($;!#(B
}
{
$B?eAG86;R$NF07BJ}8~$N4X?t(B$R_{nl}(r)$B$O(B$$B%i%2!<%k$NGfB?9`<0(B
\begin{eqnarray*}
L_p^q(x) =\frac{d^q}{dx^q}\left\{ e^x\frac{d^p}{dx^p}\left( e^{-x}x^p\right) \right\} \quad 
\end{eqnarray*}
$B$HNL;R?t(B$n,l$B!"%\!<%"H>7B(Ba_0$$B$rMQ$$$F(B
\begin{eqnarray*}
R_{nl}(r)=-\sqrt{\left( \frac{2}{na_0} \right) ^3\frac{(n-l-1)!}{2n\{(n+l)!\}^3} }e^{-\rho_n/2} \rho_n^l L_{n+l}^{2l+1}(\rho_n) 
\end{eqnarray*}
$B$HI=$5$l$k!#$^$:!"(B$L_1^1,L_2^1,L_3^3$$B$r5a$a$k$H(B,
\begin{eqnarray*}
L_1^1(r) &=&\frac{d}{dr} \left\{ e^r \frac{d}{dr} \left( e^{-r}r 
\right) \right\} \\ 
      &=& \frac{d}{dr} \left\{ e^r \left( -e^{-r}r + e^{-r} \right) \right\} \nonumber \\ 
     &=& \frac{d}{dr} \left( -r+1 \right) \\ 
     &=& -1  
\end{eqnarray*}
\begin{eqnarray*}
L_2^1(r) &=& \frac{d}{dr} \left\{ e^r \frac{d^2}{dr^2} \left( e^{-r}r^2 \right) \right\} \\ 
       &=& \frac{d}{dr} \left\{ e^r \left( e^{-r}r^2-4e^{-r}r+2e^{-r} \right) \right\} \\ 
       &=&  \frac{d}{dr} \left( r^2-4r+2 \right) \\
       &=& 2r-4  
\end{eqnarray*}
\begin{eqnarray*}
L_3^3 &=& \frac{d^3}{dr^3} \left\{ e^r \frac{d^3}{dr^3} \left( e^{-r}r^3 \right) \right\}  \\
      &=& \frac{d^3}{dr^3}\left\{ e^r \left( -e^{-r}r^3 +9e^{-r}r^2 -18e^{-r}r + 6 \right) \right\} \\
      &=&  \frac{d^3}{dr^3} \left(-r^3 +9r^2 -18r + 6 \right) \\
      &=& -6
\end{eqnarray*}
$B$H$J$k!#(B$1s,2s,2p$$B50F;$NF07BJ}8~$N4X?t$O$=$l$>$l(B$R_{10},R_{20},R_{21}$$B$G$"$j0J2<7W;;$9$k$H(B,\begin{eqnarray*}
R_{10} &=& -2 \left( \frac{1}{a_0} \right) ^{3/2}e^{-r/a_0}  L_1^1 \left( \frac{2r}{a_0} \right) \\
       &=& 2 \left( \frac{1}{a_0} \right) ^{3/2}e^{-r/a_0} 
\end{eqnarray*}
\begin{eqnarray*}
R_{20} &=& -\frac{1}{4\sqrt{2}} \left( \frac{1}{a_0} \right) ^{3/2}e^{-r/2a_0}  L_2^1 \left( \frac{r}{a_0} \right) \\
       &=& \frac{1}{2\sqrt{2}} \left( \frac{1}{a_0} \right) ^{3/2}\left( 2-\frac{r}{a_0} \right) e^{-r/2a_0} 
\end{eqnarray*}
\begin{eqnarray*}
R_{21} &=& -\frac{1}{12\sqrt{6}}\left( \frac{1}{a_0} \right) ^{3/2}\left( \frac{r}{a_0}\right) e^{-r/2a_0} L_3^3 \left( \frac{r}{a_0} \right) \\ &=& \frac{1}{2\sqrt{6}} \left( \frac{1}{a_0} \right) ^{3/2}\left( \frac{r}{a_0}\right) e^{-r/2a_0} \ \end{eqnarray*}
$B$H$J$k!#(B
}

