\documentclass{jarticle}
\title{Note.(3)}
\author{����: �g�� �՘Y}
\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{�p�^���ʂ̍���(�Â�)}
  \paragraph{�p�^���ʍ����̌`���_}
�قȂ镔����Ԃɂ����̊p�^���ʉ��Z�q${\bf J}_{1},{\bf J}_{2}$����S�p�^����${\bf J}$��
  \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}
�Œ�`����B

���P�b�g�̑I�ѕ��Ƃ��Ď��̓�ʂ肪����:
  \begin{description}
    \item[(A)]{${\bf J}_{1}^{2}, {\bf J}_{2}^{2}, J_{1z}, J_{2z}$�̓����ŗL�P�b�g$\ket{j_{1}j_{2};m_{1}m_{2}}$}
    \item[(B)]{${\bf J}^{2}, J_{z} ,{\bf J}_{1}^{2}, {\bf J}_{2}^{2}$�̓����ŗL�P�b�g$\ket{j_{1}j_{2};jm}$}
  \end{description}

�����ŁA���ϊ�
  \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}
�����藧�B�����Ō��ꂽ�ϊ��s��̗v�f$\bracket{j_{1}j_{2};m_{1}'m_{2}'}{j_{1}j_{2};jm'}$��Clebsh-Gordan�W���ł���B

Clebsh-Gordan�W���̐���:
  \begin{itemize}
    \item{$m=m_{1}'+m_{2}'$�łȂ���0}
    \item{$|j_{1}-J_{2}|\leq j \leq j_{1}+j_{2}$�łȂ���0}
    \item{Clebsh-Gordan�W���̓��j�^���s�������B���̍s��v�f�͎��ɂƂ邱�Ƃ��ł��A���̂Ƃ��͒����s��ƂȂ�B}
    \item{�K�i������ $\sum_{m_{1}'}\sum_{m_{2}'}|\bracket{j_{1}j_{2};m_{1}'m_{2}'}{j_{1}j_{2};jm}|^{2}=1$}
  \end{itemize}

note: $(J_{z}-J_{1z}-J_{2z})\ket{jm}=0$
  \paragraph{Clebsh-Gordan�W���ɑ΂���Q����}
  \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: $\mbox{��~(\ref{eq1})�ɍ�����}\bra{j_{1},j_{2};m_{1},m_{2}}J_{\pm}, m_{1}+m_{2}=m\pm 1$

$j_{1}, j_{2}$���^������ƁA$j$�Ƃ��ĈӖ��̂���l��
  \begin{displaymath}
j = j_{1} + j_{2}, j_{1} + j_{2} -1, ... , |j_{1} - j_{2}|
  \end{displaymath}
�Ɍ�����B���̒�����$j$����I�сA���̎���Clebsh-Gordan�W���̊Ԃ̊֌W��}��p���čl����B

$m_{1}'m_{2}'$-���ʂ��l����B
�����ŁA$m_{1}'$��$m_{2}'$�̘a$m_{1}'+m_{2}'$�̒l��$M$�ƕ\�L���邱�Ƃɂ���B
�Q�����ɑΉ�����}��}~\ref{fig-plus},�}~\ref{fig-minus}�Ɏ������B
  \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){�Εӏ�̓_��$m_{1}'$��$m_{2}'$�̘a��$m$.}
        \put(220,100){$\bracket{j_{1}j_{2};m_{1}m_{2}}{j_{1}j_{2};jm}$���l���A���̃u����}
        \put(210,85){�P�b�g�̕�����$J_{+} = J_{1-}^{\dagger}+J_{2-}^{\dagger}$����p������}
        \put(210,70){�ƑQ�����̍��ƌW����������.}
        \put(220,50){�Q�����Ɍ����Clebsh-Gordan�W���ł́A}
        \put(210,35){$\bracket{j_{1}j_{2};m_{1}'m_{2}'}{j_{1}j_{2};jm'}$��$m_{1}',m_{2}'$�̘a��$m'$��}
        \put(210,20){�ނ荇���Ă��邱�Ƃɒ���.}
      \end{picture}
    }
    \caption{$J_{+}$�ɂ��Clebsh-Gordan�W���̊֌W}
    \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){�Εӏ�̓_��$m_{1}'$��$m_{2}'$�̘a��$m$.}
        \put(200,100){$\bracket{j_{1}j_{2};m_{1}m_{2}}{j_{1}j_{2};jm}$���l���A���̃u����}
        \put(190,85){�P�b�g�̕�����$J_{-} = J_{1+}^{\dagger}+J_{2+}^{\dagger}$����p������}
        \put(190,70){�ƑQ�����̍��ƌW����������.}
        \put(200,50){�Q�����Ɍ����Clebsh-Gordan�W���ł́A}
        \put(190,35){$\bracket{j_{1}j_{2};m_{1}'m_{2}'}{j_{1}j_{2};jm'}$��$m_{1}',m_{2}'$�̘a��$m'$��}
        \put(190,20){�ނ荇���Ă��邱�Ƃɒ���.}
      \end{picture}
    }
    \caption{$J_{-}$�ɂ��Clebsh-Gordan�W���̊֌W}
    \label{fig-minus}
  \end{figure}

���ɁA$j_{1},j_{2},j$���^����ꂽ�Ƃ��Ɋ֌W����Clebsh-Gordan�W��$\bracket{j_{1}j_{2};m_{1}m_{2}}{j_{1}j_{2};jm}$�����ׂċ��߂���@���l����B
���̂Ƃ����e�����$m_{1},m_{2},m$�͈̔͂�
  \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}
�ł���A�����
  \begin{displaymath}
m = m_{1} + m_{2}
  \end{displaymath}
�ƂȂ�Clebsh-Gordan�W���݈̂Ӗ������B
�����$m_{1}'m_{2}'$���ʂ̐}�Ŏ����Ɛ}~\ref{fig-range}�̑��g���̒��Ƃ������ƂɂȂ�B

�܂��A�K�i����������A$m_{1}'+m_{2}'$�̒l��������Clebsh-Gordan�W��(�}~\ref{fig-range}�ł͓���ΐ���̓_)��2��a��1�ƂȂ�B

  \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{���e�����$m_{1},m_{2},m$�͈̔�}
    \label{fig-range}
  \end{figure}

�ȏ�̂��Ƃ���A�������̔C�Ӑ���������Clebsh-Gordan�W�������߂邱�Ƃ��ł���B

�ȉ��ɗ���������������B


  \subparagraph{��1: �X�s��1/2���q2�̃X�s���p�^���ʂ̍���}

��̗�Ƃ��ẮAHe���q�̓d�q��N���$(1s)^{1}(2s)^{1}$�ɂ�����C�ӂ̃X�s����ԂɑΉ�����P�b�g��W�J����2��ނ̊��P�b�g�A
����1s�O�������2s�O���ł̃X�s���ŗL�P�b�g�̒���\{$\ket{++},\ket{+-},\ket{-+},\ket{--}$\}��spin-orbit interaction���l�������ꍇ�̃G�l���M�[�ŗL�P�b�g
\{$\ket{0,0},\ket{1,+1},\ket{1,0},\ket{1,-1}$\}�̊Ԃ̊֌W���������邱�ƂȂǂ���������B

���̏��ɍl���Ă������Ƃ��ł���:
  \begin{enumerate}
    \item{�Ӗ�������$s,m$��Clebsh-Gordan�W���͐}~\ref{fig-ss}.}
    \item{�O�p�`~A���l���đQ����������.
�֎~����Ă���_�ɑ�������Clebsh-Gordan�W����0�ƂƂ�.\\
$0=\sqrt{\frac{1}{2}\cdot\frac{3}{2}-\frac{1}{2}(-\frac{1}{2})}\bracket{\frac{1}{2}\frac{1}{2};(-\frac{1}{2})(+\frac{1}{2})}{\frac{1}{2}\frac{1}{2};00}+\sqrt{\frac{1}{2}\cdot\frac{3}{2}-\frac{1}{2}(-\frac{1}{2})}\bracket{\frac{1}{2}\frac{1}{2};(+\frac{1}{2})(-\frac{1}{2})}{\frac{1}{2}\frac{1}{2};00}$.}
    \item{��������ƁA$\bracket{(-\frac{1}{2})(+\frac{1}{2})}{00}+\bracket{(+\frac{1}{2})(-\frac{1}{2})}{00} = 0$.\\
�K�i������$|\bracket{(-\frac{1}{2})(+\frac{1}{2})}{00}|^{2}+|\bracket{(+\frac{1}{2})(-\frac{1}{2})}{00}|^{2}=1$.\\
Clebsh-Gordan�W�������ɁA�܂�$\bracket{(+\frac{1}{2})(-\frac{1}{2})}{00}$�𐳂ɂƂ邱�Ƃɂ���ƁA
$\bracket{(+\frac{1}{2})(-\frac{1}{2})}{00} = 1/\sqrt{2},\quad \bracket{(-\frac{1}{2})(+\frac{1}{2})}{00}=-1/\sqrt{2}$.}
    \item{�O�p�`~B, C, D���l���đQ����������.\\
$\sqrt{1\cdot 2-1\cdot 0}\bracket{\frac{1}{2}\frac{1}{2};(-\frac{1}{2})(+\frac{1}{2})}{\frac{1}{2}\frac{1}{2};10} = \sqrt{\frac{1}{2}\cdot\frac{3}{2}-(-\frac{1}{2})\frac{1}{2}}\bracket{\frac{1}{2}\frac{1}{2};(+\frac{1}{2})(+\frac{1}{2})}{\frac{1}{2}\frac{1}{2};11}$.\\
$\sqrt{1\cdot 2-0\cdot 1}\bracket{\frac{1}{2}\frac{1}{2};(+\frac{1}{2})(+\frac{1}{2})}{\frac{1}{2}\frac{1}{2};11} = \\
\sqrt{\frac{1}{2}\cdot\frac{3}{2}-\frac{1}{2}(-\frac{1}{2})}\bracket{\frac{1}{2}\frac{1}{2};(-\frac{1}{2})(+\frac{1}{2})}{\frac{1}{2}\frac{1}{2};10}+\sqrt{\frac{1}{2}\cdot\frac{3}{2}-\frac{1}{2}(-\frac{1}{2})}\bracket{\frac{1}{2}\frac{1}{2};(+\frac{1}{2})(-\frac{1}{2})}{\frac{1}{2}\frac{1}{2};10}$.\\
$\sqrt{1\cdot 2-0\cdot(-1)}\bracket{\frac{1}{2}\frac{1}{2};(-\frac{1}{2})(-\frac{1}{2})}{\frac{1}{2}\frac{1}{2};1(-1)} = \\
\sqrt{\frac{1}{2}\cdot\frac{3}{2}-(-\frac{1}{2})\frac{1}{2}}\bracket{\frac{1}{2}\frac{1}{2};(+\frac{1}{2})(-\frac{1}{2})}{\frac{1}{2}\frac{1}{2};10}+\sqrt{\frac{1}{2}\cdot\frac{3}{2}-(-\frac{1}{2})\frac{1}{2}}\bracket{\frac{1}{2}\frac{1}{2};(-\frac{1}{2})(+\frac{1}{2})}{\frac{1}{2}\frac{1}{2};10}$.\\}
    \item{��������ƁA
$\sqrt{2}\bracket{(-\frac{1}{2})(+\frac{1}{2})}{10} = \bracket{(+\frac{1}{2})(+\frac{1}{2})}{11}$.\\
$\sqrt{2}\bracket{(+\frac{1}{2})(+\frac{1}{2})}{11} = \bracket{(-\frac{1}{2})(+\frac{1}{2})}{10}+\bracket{(+\frac{1}{2})(-\frac{1}{2})}{10}$.\\
$\sqrt{2}\bracket{(-\frac{1}{2})(-\frac{1}{2})}{1(-1)} = \bracket{(+\frac{1}{2})(-\frac{1}{2})}{10}+\bracket{(-\frac{1}{2})(+\frac{1}{2})}{10}$.\\
�K�i������$|\bracket{(+\frac{1}{2})(+\frac{1}{2})}{11}|^{2}=1$.\\
Clebsh-Gordan�W�������ɁA�܂�$\bracket{(+\frac{1}{2})(+\frac{1}{2})}{11}$�𐳂ɂƂ邱�Ƃɂ���ƁA
$\bracket{(+\frac{1}{2})(+\frac{1}{2})}{11} = 1, \quad
\bracket{(-\frac{1}{2})(+\frac{1}{2})}{10} = 1/\sqrt{2}, \quad
\bracket{(+\frac{1}{2})(-\frac{1}{2})}{10} = 1/\sqrt{2}, \quad
\bracket{(-\frac{1}{2})(-\frac{1}{2})}{1(-1)} = 1$.}
  \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�̃X�s��1/2���q�̃X�s���p�^���ʂ̍���}
    \label{fig-ss}
  \end{figure}
  \subparagraph{��2: �X�s��1/2���q1�̋O���p�^���ʂƃX�s���p�^���ʂ̍���}

��̗�Ƃ��ẮA���f���q�̓d�q�̔C�ӂ̋O���p�^����-�X�s����ԂɑΉ�����P�b�g��W�J����2��ނ̊��̃Z�b�g�A
�����X�s���������č\�������Ƃ��̃G�l���M�[�ŗL�P�b�g�ƃX�s���ŗL�P�b�g�̒��ς�spin-orbit interaction���l�������ꍇ�̃G�l���M�[�ŗL�P�b�g�̊Ԃ̊֌W���������邱�ƂȂǂ���������B

���̗�ł�$l, j, m_{j}$�Ŏw�肳�����P�b�g$\ket{l,\frac{1}{2};jm_{j}}$��$l,m_{l},m_{s}$�Ŏw�肳�����P�b�g$\ket{l,\frac{1}{2};m_{l}m_{s}}$�œW�J�����������߂�B

$j=l+\frac{1}{2}$�̂Ƃ��Ӗ��̂���Clebsh-Gordan�W���͈̔͂͐}~\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{�X�s��1/2���q�̋O���p�^���ʂƃX�s���p�^���ʂ̍���($j=l+\frac{1}{2}$)}
    \label{fig-ls-p}
  \end{figure}

  \begin{enumerate}
    \item{$m_{j}$�ɑΉ�����͍̂���.
��i�̍��ۂ���o������.\\
�O�p�`~E,F,G�ƍl���āA2���Â̑Q������M���ő�ɂȂ�܂ł���.\\
$\sqrt{j(j+1)-(m_{j}+1)m_{j}}\bracket{l\frac{1}{2};(m_{j}-\frac{1}{2})(+\frac{1}{2})}{l\frac{1}{2};jm_{j}}\\
 = \sqrt{l(l+1)-(m_{j}-\frac{1}{2})(m_{j}+\frac{1}{2})}\bracket{l\frac{1}{2};(m_{j}+\frac{1}{2})(+\frac{1}{2})}{l\frac{1}{2};j(m_{j}+1)}$.\\
$\sqrt{j(j+1)-(m_{j}+2)(m_{j}+1)}\bracket{l\frac{1}{2};(m_{j}+\frac{1}{2})(+\frac{1}{2})}{l\frac{1}{2};j(m_{j}+1)}\\
 = \sqrt{l(l+1)-(m_{j}+\frac{1}{2})(m_{j}+\frac{3}{2})}\bracket{l\frac{1}{2};(m_{j}+\frac{3}{2})(+\frac{1}{2})}{l\frac{1}{2};j(m_{j}+2)}$.\\
...\\
$\sqrt{j(j+1)-j(j-1)}\bracket{l\frac{1}{2};(l-1)(+\frac{1}{2})}{l\frac{1}{2};j(j-1)}\\
 = \sqrt{l(l+1)-(l-1)l}\bracket{l\frac{1}{2};l(+\frac{1}{2})}{l\frac{1}{2};jj}$.\\}
    \item{��i�E���̓_�ɑΉ�����$\bracket{l(+\frac{1}{2})}{jj}$�����̐���($=1$)�ɑI�ԂƁA��i��Clebsh-Gordan�W�������܂�.\\
$\bracket{(m_{j}-\frac{1}{2})(+\frac{1}{2})}{jm_{j}} = \frac{\sqrt{(l-m_{j}+\frac{1}{2})(l+m_{j}+\frac{1}{2})}\sqrt{(l-m_{j}-\frac{1}{2})(l+m_{j}+\frac{3}{2})}...\sqrt{1\cdot (l+l)}}
{\sqrt{(l+m_{j}+\frac{3}{2})(l-m_{j}+\frac{1}{2})}\sqrt{(l+m_{j}+\frac{5}{2})(l-m_{j}-\frac{1}{2})}...\sqrt{(l+l+1)\cdot 1}}
\bracket{l(+\frac{1}{2})}{jj}\\
 = \sqrt{\frac{j+m_{j}}{2j}}\bracket{l(+\frac{1}{2})}{jj} =
 \sqrt{(j+m_{j})/2j}$.\\}
    \item{�O�p�`~H���l���ĉ��i�̍��ۂɑΉ�����Clebsh-Gordan�W�������܂�.\\
$\sqrt{j(j+1)-m_{j}(m_{j}+1)}\bracket{l\frac{1}{2};(m_{j}+\frac{1}{2})(+\frac{1}{2})}{l\frac{1}{2};j(m_{j}+1)}\\
 =\sqrt{l(l+1)-(m_{j}+\frac{1}{2})(m_{j}-\frac{1}{2})}\bracket{l\frac{1}{2};(m_{j}-\frac{1}{2})(+\frac{1}{2})}{l\frac{1}{2};jm_{j}}\\
 +\sqrt{\frac{1}{2}\frac{3}{2}-\frac{1}{2}(-\frac{1}{2})}\bracket{l\frac{1}{2};(m_{j}+\frac{1}{2})(-\frac{1}{2})}{l\frac{1}{2};jm_{j}}$.\\
��������$\bracket{(m_{j}+\frac{1}{2})(-\frac{1}{2})}{jm_{j}}=\sqrt{(j-m_{j})/2j}$.\\}
  \end{enumerate}

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

�ȏ���
  \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}

�܂�$j=l-\frac{1}{2}$�̂Ƃ��Ӗ��̂���Clebsh-Gordan�W���͈̔͂͐}~\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{�X�s��1/2���q�̋O���p�^���ʂƃX�s���p�^���ʂ̍���($j=l-\frac{1}{2}$)}
    \label{fig-ls-m}
  \end{figure}

  \begin{enumerate}
    \item{$m_{j}$�ɑΉ�����͍̂���.
��i�̍��ۂ���o������.\\
$j=l+\frac{1}{2}$�̂Ƃ��Ɠ��l�ɂ���\\
$\bracket{(m_{j}-\frac{1}{2})(+\frac{1}{2})}{jm_{j}}
 = \sqrt{j-m_{j}+1}\bracket{(l-1)(+\frac{1}{2})}{jj}$.\\}
    \item{�O�p�`~Y�ɂ��āA$\sqrt{2l}\bracket{(l-1)(+\frac{1}{2})}{jj}+\bracket{l(-\frac{1}{2})}{jj}=0$.\\
�K�i������$|\bracket{(l-1)(+\frac{1}{2})}{jj}|^{2}+|\bracket{l(-\frac{1}{2})}{jj}|^{2}=1$.\\
$\bracket{l(-\frac{1}{2})}{jj}$���������ɂƂ邱�Ƃɂ���ƁA\\
$\bracket{l(-\frac{1}{2})}{jj} = \sqrt{2l/(2l+1)},\quad 
\bracket{(l-1)(+\frac{1}{2})}{jj} = -\sqrt{1/(2l+1)}$.\\
������
$\bracket{(m_{j}-\frac{1}{2})(+\frac{1}{2})}{jm_{j}} = -\sqrt{(j-m_{j}+1)/(2l+1)}$.}
    \item{�O�p�`~Z�ɂ��Č������邱�Ƃɂ��A
$\bracket{(m_{j}+\frac{1}{2})(-\frac{1}{2})}{jm_{j}} = \sqrt{(l+m+\frac{1}{2})/(2l+1)}$.}
  \end{enumerate}

�ȏ���
  \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}

$l=1$�̂Ƃ��̊��P�b�g�̎�:
  \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�W���Ɖ�]�s��}
\end{document}