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% $B:n@.G/7nF|(B : /01/12/25
% $BEE;R%a!<%k(B : g140828@mail.ecc.u-tokyo.ac.jp
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% \mondai{$B:n@.<T(B}{$B:n@.G/7nF|(B}{$BEE;R%a!<%k(B}{$BLdBjHV9f(B}{$BLdBjJ8(B}{$B2rEz(B}
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\mondai
{g140828 $BIp@n!!=c(B}
{01/12/25}
{g140828@mail.ecc.u-tokyo.ac.jp}
{9-1}
{$\ell$=0,1,2$B$N(B$Y_{\ell m} $$B$r5a$a$h!#(B}
{
$m$$B$NHO0O$O(B$m=0$,$\pm $1,$B!&!&!&(B,$\pm
$ $\ell $$B$G$"$k$N$G!"(B$\ell$=0,1,2$B$N$H$-$N(B$(\ell,m)$$B$NAH$_9g$o$;$O(B
(0,0),(1,-1),(1,0),(1,1),(2,-2),(2,-1),(2,0),(2,1),(2,2)$B$G$"$k!#(B
$B!J#9!%#2!K$N<0$h$j(B
\begin{equation}
Y_{\ell m} (\theta ,\varphi ) = \sqrt {\frac{(2\ell + 1)(\ell - \left| m
\right|)!}{4\pi (\ell + \left| m \right|)!}} P_{\ell \left| m \right|} (\cos
\theta )e^{im\varphi }
\end{equation}
$B$G$"$j!"!J#8!%#4#8!K$N<0$h$j(B
\begin{equation}
P_{\ell m} (z) = (1 - z^2)^{m / 2}\frac{d^m}{dz^m}P_\ell (z)
\end{equation}
$B$G$"$j!"!J#8!%#5#0!K$N<0$h$j(B
\begin{equation}
P_\ell (z) = \frac{1}{2^\ell \ell !}\frac{d^\ell}{dz^\ell}(z^2 - 1)^\ell
\end{equation}
$B$G$"$k!#(B
$B$h$C$F(B
\begin{eqnarray}
Y_{\ell m} (\theta ,\varphi ) = \sqrt {\frac{2\ell + 1}{4\pi }\frac{(\ell -
\left|
m \right|)!}{(\ell + \left| m \right|)!}} (1 - \cos ^2\theta )^{\left| m
\right| / 2} \nonumber \\
\times \frac{d^{\left| m \right|}}{d(\cos \theta )^{\left| m
\right|}}\frac{1}{2^\ell\ell !}\frac{d^\ell}{d(\cos \theta )^\ell}(\cos
\theta -
1)^\ell
\end{eqnarray}
$B$H$J$k!#(B
$B$3$l$K>e5-$N(B$(\ell,m)$$B$rBeF~$9$k$H!"(B
\begin{eqnarray}
Y_{00} &=& \sqrt {\frac{1}{4\pi }} ,\ Y_{1 - 1} = \sqrt {\frac{3}{8\pi }}
\sin
\theta e^{ - i\varphi }\nonumber \\
Y_{10} &=& \sqrt {\frac{3}{4\pi }} \cos \theta ,\ Y_{11} = \sqrt
{\frac{3}{8\pi }}
 \sin \theta e^{i\varphi } \nonumber \\
Y_{2 - 2} &=& \sqrt {\frac{15}{32\pi }} \sin ^2\theta e^{ - 2i\varphi } ,\
Y_{2 - 1} = \sqrt {\frac{15}{8\pi }} \sin \theta \cos \theta e^{ -
i\varphi }\nonumber \\
Y_{20} &=& \sqrt {\frac{5}{16\pi }} (3\cos ^2\theta - 1) ,\
Y_{21} = \sqrt {\frac{15}{8\pi }} \sin \theta \cos \theta e^{ - i\varphi
} \nonumber \\
Y_{22} &=& \sqrt {\frac{15}{32\pi }} \sin ^2\theta e^{2i\varphi } \nonumber
\\
\end{eqnarray}
$B$H$J$k!#(B
}




