\documentclass{jarticle}
\title{Note.(3)}
\author{ʸ��: ���� ��Ϻ}
\date{2003-03-11}
\newcommand{\ket}[1]{|#1\rangle}
\newcommand{\bra}[1]{\langle#1|}
\newcommand{\bracket}[2]{\langle#1|#2\rangle}
\newcommand{\expectation}[1]{\langle#1\rangle}
\renewcommand{\theequation}{\thesubsection.\arabic{equation}}
\begin{document}
  \maketitle
  \setcounter{section}{3}
  \setcounter{subsection}{6}
  \subsection{�ѱ�ư�̤ι���(�ĤŤ�)}
  \paragraph{�ѱ�ư�̹����η�����}
�ۤʤ���ʬ���֤ˤ�����Ĥγѱ�ư�̱黻��${\bf J}_{1},{\bf J}_{2}$�������ѱ�ư��${\bf J}$��
  \begin{displaymath}
{\bf J} \equiv {\bf J}_{1}\otimes{\bf 1} +{\bf 1}\otimes{\bf J}_{2} = {\bf J}_{1}+{\bf J}_{2}
  \end{displaymath}
��������롣

���쥱�åȤ��������Ȥ��Ƽ������̤꤬����:
  \begin{description}
    \item[(A)]{${\bf J}_{1}^{2}, {\bf J}_{2}^{2}, J_{1z}, J_{2z}$��Ʊ����ͭ���å�$\ket{j_{1}j_{2};m_{1}m_{2}}$}
    \item[(B)]{${\bf J}^{2}, J_{z} ,{\bf J}_{1}^{2}, {\bf J}_{2}^{2}$��Ʊ����ͭ���å�$\ket{j_{1}j_{2};jm}$}
  \end{description}

�����ǡ������Ѵ�
  \setcounter{equation}{32}
  \begin{equation}
    \label{eq1}
\ket{j_{1}j_{2};jm} = \sum_{m_{1}'}\sum_{m_{2}'}\ket{j_{1}j_{2};m_{1}'m_{2}'}\bracket{j_{1}j_{2};m_{1}'m_{2}'}{j_{1}j_{2};jm}
  \end{equation}
������Ω�ġ������Ǹ��줿�Ѵ����������$\bracket{j_{1}j_{2};m_{1}'m_{2}'}{j_{1}j_{2};jm'}$��Clebsh-Gordan�����Ǥ��롣

Clebsh-Gordan����������:
  \begin{itemize}
    \item{$m=m_{1}'+m_{2}'$�Ǥʤ���0}
    \item{$|j_{1}-J_{2}|\leq j \leq j_{1}+j_{2}$�Ǥʤ���0}
    \item{Clebsh-Gordan�����ϥ�˥�������Ĥ��롣���ι������Ǥϼ¤ˤȤ뤳�Ȥ��Ǥ������ΤȤ���ľ�����Ȥʤ롣}
    \item{���ʲ���� $\sum_{m_{1}'}\sum_{m_{2}'}|\bracket{j_{1}j_{2};m_{1}'m_{2}'}{j_{1}j_{2};jm}|^{2}=1$}
  \end{itemize}

note: $(J_{z}-J_{1z}-J_{2z})\ket{jm}=0$
  \paragraph{Clebsh-Gordan�������Ф���������}
  \begin{eqnarray*}
\sqrt{j(j+1)-m(m\pm 1)}\bracket{j_{1},j_{2};m_{1},m_{2}}{j_{1},j_{2};j,m\pm1}
 &=&\\
 \sqrt{j_{1}(j_{1}+1)-m_{1}(m_{1}\mp 1)}\bracket{j_{1},j_{2};m_{1}\mp 1, m_{2}}{j_{1},j_{2};j,m}
 &+&\\
 \sqrt{j_{2}(j_{2}+1)-m_{2}(m_{2}\mp 1)}\bracket{j_{1},j_{2};m_{1},m_{2}\mp 1}{j_{1},j_{2};j,m}&&\\
  \end{eqnarray*}

note: $\mbox{��~(\ref{eq1})�˺�����}\bra{j_{1},j_{2};m_{1},m_{2}}J_{\pm}, m_{1}+m_{2}=m\pm 1$

$j_{1}, j_{2}$��Ϳ������ȡ�$j$�Ȥ��ư�̣�Τ����ͤ�
  \begin{displaymath}
j = j_{1} + j_{2}, j_{1} + j_{2} -1, ... , |j_{1} - j_{2}|
  \end{displaymath}
�˸¤��롣�����椫��$j$�������ӡ����λ���Clebsh-Gordan�����δ֤δط���ޤ��Ѥ��ƹͤ��롣

$m_{1}'m_{2}'$-ʿ�̤�ͤ��롣
�����ǡ�$m_{1}'$��$m_{2}'$����$m_{1}'+m_{2}'$���ͤ�$M$��ɽ�����뤳�Ȥˤ��롣
���������б�����ޤ��~\ref{fig-plus},��~\ref{fig-minus}�˼�������
  \begin{figure}[htb]
    \fbox{
      \begin{picture}(420,200)(-10,-60)
        \put(0,100){\circle{10}}
        \put(100,0){\circle{10}}
        \put(100,100){\circle{10}}
        \put(0,100){\line(1,0){100}}
        \put(100,100){\line(0,-1){100}}
        \put(-10,110){\line(1,-1){160}}
        \put(90,110){\line(1,-1){60}}
        \put(140,-40){$M=m$}
        \put(140,60){$M=m+1$}
        \put(110,0){$(m_{1}, m_{2}-1)$}
        \put(110,100){$(m_{1}, m_{2})$}
        \put(10,100){$(m_{1}-1, m_{2})$}
        \thicklines
        \put(50,50){\vector(1,1){45}}
        \put(120,90){\vector(0,-1){80}}
        \put(90,120){\vector(-1,0){80}}
        \thinlines
        \put(50,70){$J_{+}$}
        \put(120,40){$J_{2-}^{\dagger}$}
        \put(40,125){$J_{1-}^{\dagger}$}
        \put(220,120){���վ������$m_{1}'$��$m_{2}'$���¤�$m$.}
        \put(220,100){$\bracket{j_{1}j_{2};m_{1}m_{2}}{j_{1}j_{2};jm}$��ͤ������Υ֥��}
        \put(210,85){���åȤ���ʬ��$J_{+} = J_{1-}^{\dagger}+J_{2-}^{\dagger}$����Ѥ�����}
        \put(210,70){���������ι�ȷ�����������.}
        \put(220,50){�������˸����Clebsh-Gordan�����Ǥϡ�}
        \put(210,35){$\bracket{j_{1}j_{2};m_{1}'m_{2}'}{j_{1}j_{2};jm'}$��$m_{1}',m_{2}'$���¤�$m'$��}
        \put(210,20){����äƤ��뤳�Ȥ�����.}
      \end{picture}
    }
    \caption{$J_{+}$�ˤ��Clebsh-Gordan�����δط�}
    \label{fig-plus}
  \end{figure}
  \begin{figure}[htb]
    \fbox{
      \begin{picture}(420,200)(-30,-60)
        \put(0,100){\circle{10}}
        \put(100,0){\circle{10}}
        \put(0,0){\circle{10}}
        \put(0,100){\line(0,-1){100}}
        \put(0,0){\line(1,0){100}}
        \put(-10,110){\line(1,-1){160}}
        \put(-10,10){\line(1,-1){60}}
        \put(140,-40){$M=m$}
        \put(40,-40){$M=m-1$}
        \put(110,0){$(m_{1}+1, m_{2})$}
        \put(10,0){$(m_{1}, m_{2})$}
        \put(10,100){$(m_{1}, m_{2}+1)$}
        \thicklines
        \put(50,50){\vector(-1,-1){45}}
        \put(-10,15){\vector(0,1){80}}
        \put(15,-10){\vector(1,0){80}}
        \thinlines
        \put(25,38){$J_{-}$}
        \put(-30,50){$J_{2+}^{\dagger}$}
        \put(50,-20){$J_{1+}^{\dagger}$}
        \put(200,120){���վ������$m_{1}'$��$m_{2}'$���¤�$m$.}
        \put(200,100){$\bracket{j_{1}j_{2};m_{1}m_{2}}{j_{1}j_{2};jm}$��ͤ������Υ֥��}
        \put(190,85){���åȤ���ʬ��$J_{-} = J_{1+}^{\dagger}+J_{2+}^{\dagger}$����Ѥ�����}
        \put(190,70){���������ι�ȷ�����������.}
        \put(200,50){�������˸����Clebsh-Gordan�����Ǥϡ�}
        \put(190,35){$\bracket{j_{1}j_{2};m_{1}'m_{2}'}{j_{1}j_{2};jm'}$��$m_{1}',m_{2}'$���¤�$m'$��}
        \put(190,20){����äƤ��뤳�Ȥ�����.}
      \end{picture}
    }
    \caption{$J_{-}$�ˤ��Clebsh-Gordan�����δط�}
    \label{fig-minus}
  \end{figure}

���ˡ�$j_{1},j_{2},j$��Ϳ����줿�Ȥ��˴ط�����Clebsh-Gordan����$\bracket{j_{1}j_{2};m_{1}m_{2}}{j_{1}j_{2};jm}$�򤹤٤Ƶ�����ˡ��ͤ��롣
���ΤȤ����Ƥ����$m_{1},m_{2},m$���ϰϤ�
  \begin{displaymath}
m_{1} = -j_{1}, -j_{1}+1, ... , j_{1}
  \end{displaymath}
  \begin{displaymath}
m_{2} = -j_{2}, -j_{2}+1, ... , j_{2}
  \end{displaymath}
  \begin{displaymath}
m = -j, -j+1, ... , j
  \end{displaymath}
�Ǥ��ꡢ�����
  \begin{displaymath}
m = m_{1} + m_{2}
  \end{displaymath}
�Ȥʤ�Clebsh-Gordan�����Τ߰�̣����ġ�
�����$m_{1}'m_{2}'$ʿ�̤οޤǼ����ȿ�~\ref{fig-range}������������Ȥ������Ȥˤʤ롣

�ޤ������ʲ���狼�顢$m_{1}'+m_{2}'$���ͤ�������Clebsh-Gordan����(��~\ref{fig-range}�Ǥ�Ʊ����������)��2���¤�1�Ȥʤ롣

  \begin{figure}
    \fbox{
      \begin{picture}(300,200)(-150,-100)
        \thicklines
        \put(-75,75){\line(1,0){120}}
        \put(45,75){\line(1,-1){30}}
        \put(75,45){\line(0,-1){120}}
        \put(75,-75){\line(-1,0){120}}
        \put(-45,-75){\line(-1,1){30}}
        \put(-75,-45){\line(0,1){120}}
        \thinlines
        \put(0,-85){\vector(0,1){170}}
        \put(-85,0){\vector(1,0){170}}
        \put(0,85){$m_{2}'$}
        \put(85,0){$m_{1}'$}
        \put(75,-75){\line(0,-1){15}}
        \put(75,-75){\line(1,0){15}}
        \put(-75,75){\line(0,1){15}}
        \put(-75,75){\line(-1,0){15}}
        \put(60,-95){$m_{1}'=j_{1}$}
        \put(90,-75){$m_{2}'=-j_{2}$}
        \put(-90,90){$m_{1}'=-j_{1}$}
        \put(-120,75){$m_{2}'=j_{2}$}
        \put(45,75){\line(1,-1){45}}
        \put(30,75){\line(1,-1){60}}
        \put(90,30){$M=j$}
        \put(90,15){$M=j-1$}
        \put(-45,-75){\line(-1,1){45}}
        \put(-30,-75){\line(-1,1){60}}
        \put(-130,-30){$M=-j$}
        \put(-145,-15){$M=-j+1$}
        \put(-30,75){\line(1,-1){120}}
        \put(90,-45){$M=m$}
        \put(15,30){\circle*{5}}
        \put(-35,30){$(m_{1}, m_{2})$}
      \end{picture}
    }
    \caption{���Ƥ����$m_{1},m_{2},m$���ϰ�}
    \label{fig-range}
  \end{figure}

�ʾ�Τ��Ȥ��顢�����Ĥ���Ǥ�����������Clebsh-Gordan��������뤳�Ȥ��Ǥ��롣

�ʲ�����򤤤��Ĥ���������


  \subparagraph{��1: ���ԥ�1/2γ��2�ĤΥ��ԥ�ѱ�ư�̤ι���}

������Ȥ��Ƥϡ�He���Ҥ��Ż��嵯����$(1s)^{1}(2s)^{1}$�ˤ�����Ǥ�դΥ��ԥ���֤��б����륱�åȤ�Ÿ������2����δ��쥱�åȡ�
¨��1s��ƻ�����2s��ƻ�ǤΥ��ԥ��ͭ���åȤ�ľ��\{$\ket{++},\ket{+-},\ket{-+},\ket{--}$\}��spin-orbit interaction���θ�������Υ��ͥ륮����ͭ���å�
\{$\ket{0,0},\ket{1,+1},\ket{1,0},\ket{1,-1}$\}�δ֤δط���Ƥ���뤳�Ȥʤɤ��󤲤��롣

���ν�˹ͤ��Ƥ������Ȥ��Ǥ���:
  \begin{enumerate}
    \item{��̣����$s,m$��Clebsh-Gordan�����Ͽ�~\ref{fig-ss}.}
    \item{���ѷ�~A��ͤ�����������Ĥ���.
�ػߤ���Ƥ���������������Clebsh-Gordan������0�ȤȤ�.\\
$0=\sqrt{\frac{1}{2}\cdot\frac{3}{2}-\frac{1}{2}(-\frac{1}{2})}\bracket{\frac{1}{2}\frac{1}{2};(-\frac{1}{2})(+\frac{1}{2})}{\frac{1}{2}\frac{1}{2};00}+\sqrt{\frac{1}{2}\cdot\frac{3}{2}-\frac{1}{2}(-\frac{1}{2})}\bracket{\frac{1}{2}\frac{1}{2};(+\frac{1}{2})(-\frac{1}{2})}{\frac{1}{2}\frac{1}{2};00}$.}
    \item{��������ȡ�$\bracket{(-\frac{1}{2})(+\frac{1}{2})}{00}+\bracket{(+\frac{1}{2})(-\frac{1}{2})}{00} = 0$.\\
���ʲ����$|\bracket{(-\frac{1}{2})(+\frac{1}{2})}{00}|^{2}+|\bracket{(+\frac{1}{2})(-\frac{1}{2})}{00}|^{2}=1$.\\
Clebsh-Gordan������¤ˡ��ޤ�$\bracket{(+\frac{1}{2})(-\frac{1}{2})}{00}$�����ˤȤ뤳�Ȥˤ���ȡ�
$\bracket{(+\frac{1}{2})(-\frac{1}{2})}{00} = 1/\sqrt{2},\quad \bracket{(-\frac{1}{2})(+\frac{1}{2})}{00}=-1/\sqrt{2}$.}
    \item{���ѷ�~B, C, D��ͤ�����������Ĥ���.\\
$\sqrt{1\cdot 2-1\cdot 0}\bracket{\frac{1}{2}\frac{1}{2};(-\frac{1}{2})(+\frac{1}{2})}{\frac{1}{2}\frac{1}{2};10} = \sqrt{\frac{1}{2}\cdot\frac{3}{2}-(-\frac{1}{2})\frac{1}{2}}\bracket{\frac{1}{2}\frac{1}{2};(+\frac{1}{2})(+\frac{1}{2})}{\frac{1}{2}\frac{1}{2};11}$.\\
$\sqrt{1\cdot 2-0\cdot 1}\bracket{\frac{1}{2}\frac{1}{2};(+\frac{1}{2})(+\frac{1}{2})}{\frac{1}{2}\frac{1}{2};11} = \\
\sqrt{\frac{1}{2}\cdot\frac{3}{2}-\frac{1}{2}(-\frac{1}{2})}\bracket{\frac{1}{2}\frac{1}{2};(-\frac{1}{2})(+\frac{1}{2})}{\frac{1}{2}\frac{1}{2};10}+\sqrt{\frac{1}{2}\cdot\frac{3}{2}-\frac{1}{2}(-\frac{1}{2})}\bracket{\frac{1}{2}\frac{1}{2};(+\frac{1}{2})(-\frac{1}{2})}{\frac{1}{2}\frac{1}{2};10}$.\\
$\sqrt{1\cdot 2-0\cdot(-1)}\bracket{\frac{1}{2}\frac{1}{2};(-\frac{1}{2})(-\frac{1}{2})}{\frac{1}{2}\frac{1}{2};1(-1)} = \\
\sqrt{\frac{1}{2}\cdot\frac{3}{2}-(-\frac{1}{2})\frac{1}{2}}\bracket{\frac{1}{2}\frac{1}{2};(+\frac{1}{2})(-\frac{1}{2})}{\frac{1}{2}\frac{1}{2};10}+\sqrt{\frac{1}{2}\cdot\frac{3}{2}-(-\frac{1}{2})\frac{1}{2}}\bracket{\frac{1}{2}\frac{1}{2};(-\frac{1}{2})(+\frac{1}{2})}{\frac{1}{2}\frac{1}{2};10}$.\\}
    \item{��������ȡ�
$\sqrt{2}\bracket{(-\frac{1}{2})(+\frac{1}{2})}{10} = \bracket{(+\frac{1}{2})(+\frac{1}{2})}{11}$.\\
$\sqrt{2}\bracket{(+\frac{1}{2})(+\frac{1}{2})}{11} = \bracket{(-\frac{1}{2})(+\frac{1}{2})}{10}+\bracket{(+\frac{1}{2})(-\frac{1}{2})}{10}$.\\
$\sqrt{2}\bracket{(-\frac{1}{2})(-\frac{1}{2})}{1(-1)} = \bracket{(+\frac{1}{2})(-\frac{1}{2})}{10}+\bracket{(-\frac{1}{2})(+\frac{1}{2})}{10}$.\\
���ʲ����$|\bracket{(+\frac{1}{2})(+\frac{1}{2})}{11}|^{2}=1$.\\
Clebsh-Gordan������¤ˡ��ޤ�$\bracket{(+\frac{1}{2})(+\frac{1}{2})}{11}$�����ˤȤ뤳�Ȥˤ���ȡ�
$\bracket{(+\frac{1}{2})(+\frac{1}{2})}{11} = 1, \quad
\bracket{(-\frac{1}{2})(+\frac{1}{2})}{10} = 1/\sqrt{2}, \quad
\bracket{(+\frac{1}{2})(-\frac{1}{2})}{10} = 1/\sqrt{2}, \quad
\bracket{(-\frac{1}{2})(-\frac{1}{2})}{1(-1)} = 1$.}
  \end{enumerate}
  \begin{figure}[htb]
    \fbox{
      \begin{picture}(340,170)(-165,-80)
        \put(-150,0){\vector(1,0){100}}
        \put(-100,-50){\vector(0,1){100}}
        \put(-50,0){$m_{1}'$}
        \put(-100,50){$m_{2}'$}
        \put(-80,0){$\frac{1}{2}$}
        \put(-135,0){$-\frac{1}{2}$}
        \put(-100,25){$\frac{1}{2}$}
        \put(-110,-25){$-\frac{1}{2}$}
        \put(-125,25){\circle{5}}
        \put(-75,-25){\circle{5}}
        \thicklines
        \put(-125,25){\line(1,-1){50}}
        \thinlines
        \put(-75,-25){\line(1,-1){10}}
        \put(-65,-35){$m=0$}
        \put(-125,25){\line(1,0){50}}
        \put(-75,25){\line(0,-1){50}}
        \put(-95,5){A}
        \put(-110,-75){$s=0$}
        \put(50,0){\vector(1,0){100}}
        \put(100,-50){\vector(0,1){125}}
        \put(150,0){$m_{1}'$}
        \put(100,75){$m_{2}'$}
        \put(120,0){$\frac{1}{2}$}
        \put(65,0){$-\frac{1}{2}$}
        \put(100,25){$\frac{1}{2}$}
        \put(90,-25){$-\frac{1}{2}$}
        \put(75,25){\circle{5}}
        \put(125,-25){\circle{5}}
        \put(125,25){\circle{5}}
        \put(75,-25){\circle{5}}
        \thicklines
        \put(75,25){\line(1,0){50}}
        \put(125,25){\line(0,-1){50}}
        \put(125,-25){\line(-1,0){50}}
        \put(75,-25){\line(0,1){50}}
        \thinlines
        \put(75,25){\line(1,-1){60}}
        \put(135,-35){$m=0$}
        \put(135,15){\line(-1,1){60}}
        \put(75,75){\line(0,-1){50}}
        \put(135,15){$m=1$}
        \put(75,-25){\line(-1,1){10}}
        \put(30,-15){$m=-1$}
        \put(90,40){B}
        \put(105,5){C}
        \put(90,-10){D}
        \put(90,-75){$s=1$}
      \end{picture}
    }
    \caption{2�ĤΥ��ԥ�1/2γ�ҤΥ��ԥ�ѱ�ư�̤ι���}
    \label{fig-ss}
  \end{figure}
  \subparagraph{��2: ���ԥ�1/2γ��1�Ĥε�ƻ�ѱ�ư�̤ȥ��ԥ�ѱ�ư�̤ι���}

������Ȥ��Ƥϡ����Ǹ��Ҥ��ŻҤ�Ǥ�դε�ƻ�ѱ�ư��-���ԥ���֤��б����륱�åȤ�Ÿ������2����δ���Υ��åȡ�
¨�����ԥ������ƹ��������Ȥ��Υ��ͥ륮����ͭ���åȤȥ��ԥ��ͭ���åȤ�ľ�Ѥ�spin-orbit interaction���θ�������Υ��ͥ륮����ͭ���åȤδ֤δط���Ƥ���뤳�Ȥʤɤ��󤲤��롣

������Ǥ�$l, j, m_{j}$�ǻ��ꤵ�����쥱�å�$\ket{l,\frac{1}{2};jm_{j}}$��$l,m_{l},m_{s}$�ǻ��ꤵ�����쥱�å�$\ket{l,\frac{1}{2};m_{l}m_{s}}$��Ÿ������������롣

$j=l+\frac{1}{2}$�ΤȤ���̣�Τ���Clebsh-Gordan�������ϰϤϿ�~\ref{fig-ls-p}.

  \begin{figure}[htb]
    \fbox{
      \begin{picture}(320,150)(-35,-65)
        \put(0,25){\circle{5}}
        \put(50,25){\circle*{5}}
        \multiput(100,25)(50,0){3}{\circle{5}}
        \multiput(0,-25)(50,0){2}{\circle{5}}
        \put(100,-25){\circle*{5}}
        \multiput(150,-25)(50,0){2}{\circle{5}}
        \put(-25,0){\vector(1,0){250}}
        \put(0,-50){\vector(0,1){100}}
        \put(225,0){$m_{l}'$}
        \put(0,50){$m_{s}'$}
        \thicklines
        \put(-25,25){\line(1,0){225}}
        \put(200,25){\line(0,-1){50}}
        \put(200,-25){\line(-1,0){225}}
        \thinlines
        \put(200,-25){\line(0,-1){15}}
        \put(200,-40){$m_{l}'=l$}
        \put(50,25){\line(1,-1){65}}
        \put(115,-50){$M=m_{j}$}
        \put(50,25){\line(0,1){50}}
        \put(50,75){\line(1,-1){50}}
        \put(100,75){\line(0,-1){100}}
        \put(100,75){\line(1,-1){50}}
        \put(150,75){\line(0,-1){50}}
        \put(150,75){\line(1,-1){65}}
        \put(215,10){$M=j=l+\frac{1}{2}$}
        \put(-10,25){$\frac{1}{2}$}
        \put(-20,-25){$-\frac{1}{2}$}
        \put(65,40){E}
        \put(115,40){F}
        \put(165,40){G}
        \put(80,5){H}
      \end{picture}
    }
    \caption{���ԥ�1/2γ�Ҥε�ƻ�ѱ�ư�̤ȥ��ԥ�ѱ�ư�̤ι���($j=l+\frac{1}{2}$)}
    \label{fig-ls-p}
  \end{figure}

  \begin{enumerate}
    \item{$m_{j}$���б�����ΤϹ���.
���ʤι��ݤ����ȯ����.\\
���ѷ�~E,F,G�ȹͤ��ơ�2��ŤĤ���������M������ˤʤ�ޤǤĤ���.\\
$\sqrt{j(j+1)-(m_{j}+1)m_{j}}\bracket{l\frac{1}{2};(m_{j}-\frac{1}{2})(+\frac{1}{2})}{l\frac{1}{2};jm_{j}}\\
 = \sqrt{l(l+1)-(m_{j}-\frac{1}{2})(m_{j}+\frac{1}{2})}\bracket{l\frac{1}{2};(m_{j}+\frac{1}{2})(+\frac{1}{2})}{l\frac{1}{2};j(m_{j}+1)}$.\\
$\sqrt{j(j+1)-(m_{j}+2)(m_{j}+1)}\bracket{l\frac{1}{2};(m_{j}+\frac{1}{2})(+\frac{1}{2})}{l\frac{1}{2};j(m_{j}+1)}\\
 = \sqrt{l(l+1)-(m_{j}+\frac{1}{2})(m_{j}+\frac{3}{2})}\bracket{l\frac{1}{2};(m_{j}+\frac{3}{2})(+\frac{1}{2})}{l\frac{1}{2};j(m_{j}+2)}$.\\
...\\
$\sqrt{j(j+1)-j(j-1)}\bracket{l\frac{1}{2};(l-1)(+\frac{1}{2})}{l\frac{1}{2};j(j-1)}\\
 = \sqrt{l(l+1)-(l-1)l}\bracket{l\frac{1}{2};l(+\frac{1}{2})}{l\frac{1}{2};jj}$.\\}
    \item{���ʱ����������б�����$\bracket{l(+\frac{1}{2})}{jj}$��¤�����($=1$)�����֤ȡ����ʤ�Clebsh-Gordan��������ޤ�.\\
$\bracket{(m_{j}-\frac{1}{2})(+\frac{1}{2})}{jm_{j}} = \frac{\sqrt{(l-m_{j}+\frac{1}{2})(l+m_{j}+\frac{1}{2})}\sqrt{(l-m_{j}-\frac{1}{2})(l+m_{j}+\frac{3}{2})}...\sqrt{1\cdot (l+l)}}
{\sqrt{(l+m_{j}+\frac{3}{2})(l-m_{j}+\frac{1}{2})}\sqrt{(l+m_{j}+\frac{5}{2})(l-m_{j}-\frac{1}{2})}...\sqrt{(l+l+1)\cdot 1}}
\bracket{l(+\frac{1}{2})}{jj}\\
 = \sqrt{\frac{j+m_{j}}{2j}}\bracket{l(+\frac{1}{2})}{jj} =
 \sqrt{(j+m_{j})/2j}$.\\}
    \item{���ѷ�~H��ͤ��Ʋ��ʤι��ݤ��б�����Clebsh-Gordan�������ޤ�.\\
$\sqrt{j(j+1)-m_{j}(m_{j}+1)}\bracket{l\frac{1}{2};(m_{j}+\frac{1}{2})(+\frac{1}{2})}{l\frac{1}{2};j(m_{j}+1)}\\
 =\sqrt{l(l+1)-(m_{j}+\frac{1}{2})(m_{j}-\frac{1}{2})}\bracket{l\frac{1}{2};(m_{j}-\frac{1}{2})(+\frac{1}{2})}{l\frac{1}{2};jm_{j}}\\
 +\sqrt{\frac{1}{2}\frac{3}{2}-\frac{1}{2}(-\frac{1}{2})}\bracket{l\frac{1}{2};(m_{j}+\frac{1}{2})(-\frac{1}{2})}{l\frac{1}{2};jm_{j}}$.\\
��������$\bracket{(m_{j}+\frac{1}{2})(-\frac{1}{2})}{jm_{j}}=\sqrt{(j-m_{j})/2j}$.\\}
  \end{enumerate}

note: $j(j+1)-m(m+1) = (j-m)(j+m+1), j(j+1)-m(m-1)=(j+m)(j-m+1)$

�ʾ���
  \begin{displaymath}
\ket{j=l+\frac{1}{2},m_{j}} = 
\sqrt{\frac{l+m_{j}+\frac{1}{2}}{2l+1}}\ket{m_{j}-\frac{1}{2},+}
+
\sqrt{\frac{l-m_{j}+\frac{1}{2}}{2l+1}}\ket{m_{j}+\frac{1}{2},-}
  \end{displaymath}

�ޤ�$j=l-\frac{1}{2}$�ΤȤ���̣�Τ���Clebsh-Gordan�������ϰϤϿ�~\ref{fig-ls-m}.

  \begin{figure}[htb]
    \fbox{
      \begin{picture}(370,150)(-35,-65)
        \put(0,25){\circle{5}}
        \put(50,25){\circle*{5}}
        \multiput(100,25)(50,0){3}{\circle{5}}
        \multiput(0,-25)(50,0){2}{\circle{5}}
        \put(100,-25){\circle*{5}}
        \multiput(150,-25)(50,0){3}{\circle{5}}
        \put(-25,0){\vector(1,0){300}}
        \put(0,-50){\vector(0,1){100}}
        \put(275,0){$m_{l}'$}
        \put(0,50){$m_{s}'$}
        \thicklines
        \put(-25,25){\line(1,0){225}}
        \put(200,25){\line(1,-1){50}}
        \put(250,-25){\line(-1,0){275}}
        \thinlines
        \put(250,25){\line(-1,0){50}}
        \put(250,-40){\line(0,1){65}}
        \put(230,-50){$m_{l}'=l$}
        \put(50,25){\line(1,-1){65}}
        \put(115,-50){$M=m_{j}$}
        \put(50,25){\line(0,1){50}}
        \put(50,75){\line(1,-1){50}}
        \put(100,75){\line(0,-1){100}}
        \put(100,75){\line(1,-1){50}}
        \put(150,75){\line(0,-1){50}}
        \put(150,75){\line(1,-1){50}}
        \put(250,-25){\line(1,-1){15}}
        \put(265,-40){$M=j=l-\frac{1}{2}$}
        \put(-10,25){$\frac{1}{2}$}
        \put(-20,-25){$-\frac{1}{2}$}
        \put(65,40){V}
        \put(115,40){W}
        \put(165,40){X}
        \put(230,5){Y}
        \put(80,5){Z}
      \end{picture}
    }
    \caption{���ԥ�1/2γ�Ҥε�ƻ�ѱ�ư�̤ȥ��ԥ�ѱ�ư�̤ι���($j=l-\frac{1}{2}$)}
    \label{fig-ls-m}
  \end{figure}

  \begin{enumerate}
    \item{$m_{j}$���б�����ΤϹ���.
���ʤι��ݤ����ȯ����.\\
$j=l+\frac{1}{2}$�ΤȤ���Ʊ�ͤˤ���\\
$\bracket{(m_{j}-\frac{1}{2})(+\frac{1}{2})}{jm_{j}}
 = \sqrt{j-m_{j}+1}\bracket{(l-1)(+\frac{1}{2})}{jj}$.\\}
    \item{���ѷ�~Y�ˤĤ��ơ�$\sqrt{2l}\bracket{(l-1)(+\frac{1}{2})}{jj}+\bracket{l(-\frac{1}{2})}{jj}=0$.\\
���ʲ����$|\bracket{(l-1)(+\frac{1}{2})}{jj}|^{2}+|\bracket{l(-\frac{1}{2})}{jj}|^{2}=1$.\\
$\bracket{l(-\frac{1}{2})}{jj}$��¤������ˤȤ뤳�Ȥˤ���ȡ�\\
$\bracket{l(-\frac{1}{2})}{jj} = \sqrt{2l/(2l+1)},\quad 
\bracket{(l-1)(+\frac{1}{2})}{jj} = -\sqrt{1/(2l+1)}$.\\
������
$\bracket{(m_{j}-\frac{1}{2})(+\frac{1}{2})}{jm_{j}} = -\sqrt{(j-m_{j}+1)/(2l+1)}$.}
    \item{���ѷ�~Z�ˤĤ��Ƹ�Ƥ���뤳�Ȥˤ�ꡢ
$\bracket{(m_{j}+\frac{1}{2})(-\frac{1}{2})}{jm_{j}} = \sqrt{(l+m+\frac{1}{2})/(2l+1)}$.}
  \end{enumerate}

�ʾ���
  \begin{displaymath}
\ket{j=l-\frac{1}{2},m_{j}} = 
-\sqrt{\frac{l-m_{j}+\frac{1}{2}}{2l+1}}\ket{m_{j}-\frac{1}{2},+}
+
\sqrt{\frac{l+m_{j}+\frac{1}{2}}{2l+1}}\ket{m_{j}+\frac{1}{2},-}
  \end{displaymath}

$l=1$�ΤȤ��δ��쥱�åȤμ�:
  \begin{eqnarray*}
\ket{\frac{3}{2},\frac{3}{2}} &=& \ket{1,+}\\
\ket{\frac{3}{2},\frac{1}{2}} &=& \sqrt{\frac{2}{3}}\ket{0,+} + \sqrt{\frac{1}{3}}\ket{1,-}\\
\ket{\frac{3}{2},-\frac{1}{2}} &=& \sqrt{\frac{1}{3}}\ket{-1,+} + \sqrt{\frac{2}{3}}\ket{0,-}\\
\ket{\frac{3}{2},-\frac{3}{2}} &=& \ket{-1,-}\\
\ket{\frac{1}{2},\frac{1}{2}} &=& -\sqrt{\frac{1}{3}}\ket{0,+} + \sqrt{\frac{2}{3}}\ket{1,-}\\
\ket{\frac{1}{2},-\frac{1}{2}} &=& -\sqrt{\frac{2}{3}}\ket{-1,+} + \sqrt{\frac{1}{3}}\ket{0,-}\\
  \end{eqnarray*}


  \paragraph{Clebsh-Gordan�����Ȳ�ž����}
\end{document}