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一般の$m$の場合
\begin{eqnarray}
|l,\frac{1}{2},j=l+\frac{1}{2},m\rangle&=&\left(\frac{l+m+\frac{1}{2}}{2l+1}\right)^{\frac{1}{2}}Y_{l,m-\frac{1}{2}}\chi_{\frac{1}{2},\frac{1}{2}}+\left(\frac{l-m+\frac{1}{2}}{2l+1}\right)^{\frac{1}{2}}Y_{l,m+\frac{1}{2}}\chi_{\frac{1}{2},-\frac{1}{2}}\\
&=&\left(\frac{j+m}{2j}\right)^{\frac{1}{2}}Y_{l,m-\frac{1}{2}}\chi_{\frac{1}{2},\frac{1}{2}}+\left(\frac{j-m}{2j}\right)^{\frac{1}{2}}Y_{l,m+\frac{1}{2}}\chi_{\frac{1}{2},-\frac{1}{2}}
\end{eqnarray}

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