%<PLAINTEXT>
%$BNL;RJ*M}3X(B : $B1i=,LdBj(B9-2$B2rEz(B
%$B:n@.<T(B     : 250237G $BJ?Ln2m5,(B
%$B:n@.G/7nF|(B :  2002/11/26
%$BEE;R%a!<%k(B : g250237@mail.ecc.u-tokyo.ac.jp
  \mondai {250237G $BJ?Ln2m5,(B}
{2002/11/26}
{g250237@mail.ecc.u-tokyo.ac.jp}
{9-2} {1$B$G5a$a$?(B $Y_{lm}$ $B$OJ#AG?t$G$"$k!#$3$3$G(B $Y_{lm} \pm Y_{l-m}$ $B7?$N4X(B
$B?t$r9M$($l$P<B?t$N4X?t$K$J$k$3$H$r<($;!#5,3JD>8r2=$9$k$h$&$K78?t$bDj$a$h!#(B}
{
(9.3)$B<0$h$j!"(B
\begin{equation}
 \int^{\pi}_{0} \sin\theta d \theta \int^{2\pi}_{0} d\varphi Y^\ast_{lm}
Y_{{l}^\prime {m}^\prime } = \delta_{l^\prime {l}} \delta_{m^\prime {m}}
\end{equation}           
$B$@$+$i!"(B$Y_{lm} \pm Y_{l-m}$$B$r5,3JD>8r2=$9$k0Y$K>l9g$K1~$8$F!"(B
$\frac{1}{\sqrt2}$(or $\frac{1}{{\sqrt2}i}$) $BG\$9$k!#$^$?!"(B(8.40)$B<0$N6K:BI8(B
$BI=<((B
\[
\left\{
 \begin{array}{l}
x=r\sin\theta \cos\varphi \\
y=r\sin\theta \sin\varphi  \\
z=r\cos\theta
\end{array}
\right.
\]
$B$rMQ$$$k!#(B$m \neq 0 $$B$N;~!"LdBj(B(9.1)$B$+$i(B
\begin{equation}
Y_{11} + Y_{1 -1 }=
\sqrt{\frac{3}{8\pi}} \sin \theta e^{i \varphi} + \sqrt{\frac{3}{8\pi}} \sin
\theta e^{-i \varphi} = 2\sqrt{\frac{3}{8\pi}} \sin \theta \cos \varphi
= \sqrt{\frac{3}{2\pi}} \frac{x}{r} ,  \end{equation}
$B$3$l$r5,3J2=$7$F(B${\displaystyle{\sqrt{\frac{3}{4\pi}}} \frac{x}{r}}$$B!#(B
\begin{equation}
 Y_{11}-Y_{1-1}= \sqrt{\frac{3}{8\pi}} \sin\theta
 e^{i\varphi} -\sqrt{\frac{3}{8\pi}}\sin \theta
e^{-i\varphi}=i\sqrt{\frac{3}{2\pi}} \sin \theta \sin \varphi   =
i\sqrt{\frac{3}{2\pi}} \frac{y}{r} ,  \end{equation}
$B$3$l$r5,3J2=$7$F(B ${\displaystyle{\sqrt{\frac{3}{4\pi}} \frac{y}{r}}}$$B!#(B
\begin{eqnarray}
Y_{21}+Y_{2-1} &=& \sqrt{\frac{15}{8\pi}} \sin\theta \cos\theta e^{i\varphi}
+\sqrt{\frac{15}{8\pi}}\sin \theta \cos \theta e^{-i\varphi}  \nonumber \\
&=&  2\sqrt{\frac{15}{8\pi}}\sin\theta \cos\theta \cos\varphi   \nonumber 
\\
&=& \sqrt{\frac{15}{2\pi}} \frac{xz}{r^2} ,
\end{eqnarray}
$B$3$l$r5,3J2=$7$F(B${\displaystyle{\sqrt{\frac{15}{4\pi}}} \frac{xz}{r^2}}$$B!#(B

\begin{eqnarray}
Y_{21}-Y_{2-1} &=& \sqrt{\frac{15}{8\pi}} \sin\theta \cos\theta
e^{i\varphi}-\sqrt{\frac{15}{8\pi}} \sin\theta \cos\theta e^{-i\varphi}
\nonumber \\
&=& 2i\sqrt{\frac{15}{8\pi}}\sin \theta \cos \theta \sin \varphi   \nonumber
\\
&=& i\sqrt{\frac{15}{2\pi}} \frac{yz}{r^2},
\end{eqnarray}
$B$3$l$r5,3J2=$7$F(B ${\displaystyle{\sqrt{\frac{15}{4\pi}} \frac{yz}{r^2}}}$$B!#(B
\begin{eqnarray}
Y_{22} +Y_{2-2} &=&  \sqrt{\frac{15}{32\pi}} \displaystyle{\left( {\sin^2}
\theta e^{2i\varphi}+
{\sin^2} \theta e^{-2i\varphi} \right)}  \nonumber \\
&=& \sqrt{\frac{15}{8\pi}} {\sin^2 }\theta {({\cos ^2}\varphi -{\sin^2}
\varphi) }  \nonumber \\
 &=& \sqrt{\frac{15}{8\pi}} \displaystyle{\left( \left( \frac{x}{r} \right)
{}^2-\left( \frac{y}{r} \right) {}^2 \right) },
\end{eqnarray}
$B$3$l$r5,3J2=$7$F(B 
${\displaystyle{\sqrt{\frac{15}{16\pi}}}\frac{x^2-y^2}{r^2}}$$B!#(B
\begin{equation}
Y_{22}-Y_{2-2}=2i\sqrt{\frac{15}{32\pi}} {\sin^2} \theta \sin 2\varphi
=2i\sqrt{\frac{15}{8\pi}}{\sin ^2}\theta \sin \varphi \cos \varphi
=i\sqrt{\frac{15}{2\pi}} \frac{xy}{r^2},
\end{equation}
$B$3$l$r5,3J2=$7$F(B${\displaystyle{\sqrt{\frac{15}{4\pi}}}\frac{xy}{r^2}
}$ $B!#(B
$B$^$?!"(B\begin{equation}
 Y_{10}=\sqrt{\frac{3}{4\pi}}\cos
\theta=\sqrt{\frac{3}{4\pi}}\frac{z}{r},Y_{20}=\sqrt{\frac{5}{16\pi}}(3
{\cos^2 \theta}-1)=\sqrt{\frac{5}{4\pi}}\frac{3z^2-r^2}{2r^2} 
\end{equation}
$B$h$j!"(B$Y_{lm} \pm Y_{l-m}$$B7?$N4X?t$r9M$($l$P!"<B?t$N4X?t$K$J$k$3$H$,<($5$l(B
$B$?!#(B
}

