\documentclass{jarticle}
\title{Note.(2)}
\author{����: �g�� �՘Y}
\date{2003-03-07}
\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}{4}
  \subsection{�p�^���ʂ̌ŗL�l�ƌŗL���}
  \paragraph{�����֌W�Ƃ͂������Z�q}
  \begin{equation}
{\bf J}^{2} \equiv J_{x}^{2} + J_{y}^{2} + J_{z}^{2}
  \end{equation}
�Œ�`�����${\bf J}^{2}$��$J_{k}$�ƌ�������:
  \begin{equation}
[{\bf J}^{2}, J_{k}] = 0
  \end{equation}

${\bf J}^{2}$��$J_{z}$����������Ƃ���ƁA�����̓����ŗL�P�b�g$\ket{a,b}$�������Ď��������藧�B
  \setcounter{equation}{3}
  \begin{equation}
{\bf J}^{2}\ket{a,b} = a\ket{a,b}
  \end{equation}
  \begin{displaymath}
J_{z}\ket{a,b} = b\ket{a,b}
  \end{displaymath}

�����ŁA�͂������Z�q
  \begin{equation}
J_{\pm} \equiv  J_{x} \pm i J_{y}
  \end{equation}
���`����ƁA$J_{\pm}$��$\ket{a,b}$�Ɏ��̂悤�ɍ�p����:
  \setcounter{equation}{11}
  \begin{equation}
J_{\pm}\ket{a,b} = c_{\pm}\ket{a, b \pm \hbar}
  \end{equation}

$c_{\pm}$�ɂ��Ă͍s��v�f�����߂�Ƃ��ɋ��߂�B

note: $[J_{x}, J_{y}] = i\hbar J_{z}, [J_{z}, J_{\pm}] = \pm \hbar J_{\pm}, J_{z}J_{\pm}\ket{a,b} = (b+\hbar)J_{\pm}\ket{a,b}$
  \paragraph{${\bf J}^{2}$�����$J_{z}$�̌ŗL�l}
  \begin{equation}
a \geq b^{2}
  \end{equation}
�ł��邱�Ƃ��A${\bf J}^{2}-J_{z}^{2}$�̊��Ғl���l���邱�Ƃł킩��B

note: $J_{+}J_{-} + J_{-}J_{+} = 2(J_{x}^{2} + J_{y}^{2}), J_{-} = J_{+}^{\dagger}, \mbox{�m����}\geq 0$

������Ab�ɂ͏㉺�������݂��邱�Ƃ��킩��̂ŁA�����𖞂����K�v������B
  \setcounter{equation}{16}
  \begin{equation}
J_{+}\ket{a, b_{max}} = 0
  \end{equation}
  \setcounter{equation}{22}
  \begin{equation}
J_{-}\ket{a, b_{min}} = 0
  \end{equation}

�����ŁA
  \begin{displaymath}
J_{-}J_{+}\ket{a, b_{max}} = (a-b_{max}^{2}-\hbar b_{max})\ket{a, b_{max}} = 0
  \end{displaymath}
  \begin{displaymath}
J_{+}J_{-}\ket{a, b_{mix}} = (a-b_{min}^{2}+\hbar b_{mix})\ket{a, b_{mix}} = 0
  \end{displaymath}
�ł��邩��A
  \begin{displaymath}
a = b_{max}(b_{max} + \hbar) =  (-b_{min})((-b_{mix}) + \hbar)
  \end{displaymath}
����āA
  \setcounter{equation}{25}
  \begin{equation}
b_{max} = - b_{min}
  \end{equation}
�Ɠ�����B

note: $J_{+}J_{-} = {\bf J}^{2}-J_{z}^{2}+\hbar J_{z}$

������A$\ket{a, -b_{max}}$��$J_{+}$��n���p�������$\ket{a, b_{max}}$�ɂȂ�Ƃ���Ƃ��A
  \setcounter{equation}{28}
  \begin{equation}
b_{max} = \frac{n\hbar}{2}
  \end{equation}
�ł���B�����ŁA
  \begin{equation}
j \equiv \frac{b_{max}}{\hbar}
  \end{equation}
  \setcounter{equation}{31}
  \begin{equation}
b \equiv m\hbar
  \end{equation}
�ƒ�`����ƁA
  \setcounter{equation}{29}
  \begin{equation}
j = \frac{n}{2}
  \end{equation}
  \begin{equation}
a = \hbar^{2}j(j+1)
  \end{equation}
  \setcounter{equation}{32}
  \begin{equation}
m = -j, -j+1, ... , j-1, j
  \end{equation}
�ƂȂ�B�䂦�ɁA
${\bf J}^{2}$��$J_{z}$�̓����ŗL�P�b�g��$\ket{j,m}$�ł���킷�Ƃ��ŗL�l��������
  \begin{equation}
{\bf J}^{2}\ket{j,m} = j(j+1)\hbar^{2}\ket{j,m}
  \end{equation}
  \begin{displaymath}
J_{z}\ket{j,m} = m\hbar\ket{j,m}
  \end{displaymath}
�ƂȂ�B
  \paragraph{�p�^���ʉ��Z�q�̍s��v�f}
  \begin{equation}
\bra{j',m'}{\bf J}^{2}\ket{j,m} = j(j+1)\hbar^{2}\delta_{j'j}\delta_{m'm}
  \end{equation}
  \begin{displaymath}
\bra{j',m'}J_{z}\ket{j,m} = m\hbar\delta_{j'j}\delta_{m'm}
  \end{displaymath}
  \setcounter{equation}{40}
  \begin{equation}
\bra{j',m'}J_{\pm}\ket{j,m} = \sqrt{j(j+1)-m(m \pm 1)}\hbar\delta_{j'j}\delta_{m'm \pm 1}
  \end{equation}

note: $J_{\pm}\ket{j,m} = c_{\pm}\ket{j,m\pm 1}, J_{\mp}J_{\pm} = {\bf J}^{2}-J_{z}^{2}\mp\hbar J_{z}, \mbox{�m����} = |c_{\pm}|^{2} = \{j(j+1)-m(m\pm 1)\}\hbar^{2}$
  \paragraph{��]���Z�q�̕\��}
��]���Z�q${\cal D}(R)$�̍s��v�f�́A��]$R$��${\bf n}$��$\phi$�Ŏw�肳���Ƃ��A
  \begin{equation}
{\cal D}^{(j)}_{m'm}(R) = \bra{j, m'}\exp(\frac{-i{\bf J}\cdot{\bf n}\phi}{\hbar})\ket{j, m}
  \end{equation}
  \subsection{�O���p�^����}
  \setcounter{equation}{0}
�O���p�^����${\bf L}$��
  \begin{equation}
{\bf L} \equiv {\bf x} \times {\bf p}
  \end{equation}
�ƒ�`����B�ꗱ�q�ɂ��ē������R�x�𖳎��ł���ꍇ��${\bf L}$��${\bf J}$�Ɉ�v���邱�Ƃ��ȉ��Ŏ����B
  \paragraph{��]�̐������Z�q�Ƃ��Ă̋O���p�^����}
${\bf L}$�́A
  \begin{itemize}
    \item{�p�^���ʂ̌����֌W $[L_{x}, L_{y}] = i\hbar L_{z}$}
    \item{��]�̐��� $({\bf 1} - iL_{z}\delta\phi/\hbar)\ket{x',y',z'} = \ket{x'-y'\delta\phi, y'+x'\delta\phi, z'}$}
  \end{itemize}
�𖞑�����B

note: $[x_{i}, x_{j}] = 0, [p_{i}, p_{j}] = 0, [x_{i}, p_{j}] = i\hbar\delta_{ij}, L_{z}=xp_{y}-yp_{x}, {\cal T}(d{\bf x'})={\bf 1}-i{\bf p}d{\bf x'}/\hbar$

$\bracket{x',y',z'}{\alpha}$�ŗ^������g���֐������ꗱ�q��z���܂��Ŗ�������]�����邱�Ƃ��l����B
  \begin{eqnarray*}
\bracket{x',y',z'}{\alpha} 
\rightarrow
\bra{x',y',z'}({\bf 1} - \frac{iL_{z}\delta\phi}{\hbar})\ket{\alpha}
&=&\bracket{x'+y'\delta\phi,y'-x'\delta\phi,z'}{\alpha}\\
&=&\bracket{r, \theta, \phi - \delta\phi}{\alpha}\\
&=&\bracket{r, \theta, \phi}{\alpha}-\delta\phi\frac{\partial}{\partial\phi}\bracket{r, \theta, \phi}{\alpha}
  \end{eqnarray*}
���A
  \setcounter{equation}{8}
  \begin{equation}
\bra{{\bf x'}}L_{z}\ket{\alpha} = -i\hbar\frac{\partial}{\partial\phi}\bracket{{\bf x'}}{\alpha}
  \end{equation}
��������B

$L_{x}, L_{y}, L_{\pm}$�ɂ��Ă����l�̌������s�����Ƃɂ��A
  \begin{displaymath}
\bra{{\bf x'}}{\bf L}^{2}\ket{\alpha} = -\hbar^{2}\Lambda^{2}\bracket{{\bf x'}}{\alpha}
  \end{displaymath}
�Ƃ�����B

note:
  \begin{displaymath}
{\bf L}^{2} = L_{z}^{2} + \frac{1}{2}(L_{+}L_{-}-L_{-}L_{+})
  \end{displaymath}
  \begin{displaymath}
\Lambda^{2} \psi = \frac{1}{\sin^{2}\theta}\frac{\partial^{2}\psi}{\partial\phi^{2}}+\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}(\sin\theta\frac{\partial\psi}{\partial\theta})
  \end{displaymath}
  \begin{displaymath}
\nabla^{2} \psi = \frac{1}{r}\frac{\partial^{2}(r\psi)}{\partial r^{2}} + \frac{1}{r^{2}}\Lambda^{2}\psi
  \end{displaymath}

  \paragraph{���ʒ��a�֐�}
���Ώ̂̃|�e���V�����̂��ƂŃX�s���������Ȃ���̗��q���l����B
�G�l���M�[�ŗL�֐���
  \setcounter{equation}{21}
  \begin{equation}
\bracket{{\bf x'}}{n, l, m} = R_{nl}(r)Y_{l}^{m}(\theta, \phi)
  \end{equation}

note: $H$�����Ώ̂ȂƂ�$H$��$L_{z}, {\bf L}^{2}$�ƌ����B�G�l���M�[�ŗL�P�b�g���O���p�^���ʂ̌ŗL�P�b�g�B

�p�x�ˑ����ɂ��čl����B
�����ŗL�P�b�g$\ket{{\bf n}}$���`���A�p�x�ˑ������𕪗��������̋��ʒ��a�֐����l����B
  \begin{equation}
\bracket{{\bf n}}{l,m} = Y_{l}^{m}(\theta, \phi) = Y_{l}^{m}({\bf n})
  \end{equation}

�O���p�^���ʂ̌ŗL�P�b�g$\ket{l,m}$���܂ފ֌W������Ή����鋅�ʒ��a�֐����܂ފ֌W����������B

��1:
  \begin{equation}
L_{z}\ket{l,m} = m\hbar\ket{l, m}
  \end{equation}
�����
  \setcounter{equation}{25}
  \begin{equation}
-i\hbar\frac{\partial}{\partial\phi}Y_{l}^{m}(\theta, \phi) = m\hbar Y_{l}^{m}(\theta, \phi)
  \end{equation}
��������B$Y_{l}^{m}(\theta, \phi)$��$\phi$�ˑ�������$e^{im\phi}$�̌`�ɂȂ邱�Ƃ��킩��B

��2:
  \begin{equation}
{\bf L}^{2}\ket{l,m} = l(l+1)\hbar^{2}\ket{l, m}
  \end{equation}
�����
  \begin{equation}
\frac{1}{\sin^{2}\theta}\frac{\partial^{2}}{\partial\phi^{2}}+\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}(\sin\theta\frac{\partial}{\partial\theta} + l(l+1))Y_{l}^{m}=0
  \end{equation}
��������B

note: �܂�������$\bra{{\bf n}}$��������B

��3:
  \begin{equation}
\bracket{l',m'}{l,m} = \delta_{ll'}\delta_{mm'}
  \end{equation}
�����
  \begin{equation}
\int^{2\pi}_{0}d\phi\int^{1}_{-1}d(\cos\theta)Y_{l'}^{m'*}(\theta, \phi)Y_{l}^{m}(\theta, \phi) = \delta_{ll'}\delta_{mm'}
  \end{equation}
��������B

note: �܂��^�񒆂�$\int d\Omega_{{\bf n}}\ket{{\bf n}}\bra{{\bf n}} = 1$��}������B

  \setcounter{equation}{36}
  \begin{equation}
Y_{l}^{m}(\theta, \phi) = \frac{(-1)^{l}}{2^{l}l!}\sqrt{\frac{(2l+1)(l+m)!}{4\pi(l-m)!}}e^{im\phi}\frac{1}{\sin^{m}\theta}\frac{d^{(l-m)}}{d(\cos\theta)^{(l-m)}}(\sin\theta)^{2l}\quad(m \geq 0)
  \end{equation}
  \begin{equation}
Y_{l}^{-m}(\theta, \phi) = (-1)^{m}(Y_{l}^{m}(\theta, \phi))^{*}
  \end{equation}
  \paragraph{��]�s��Ƃ��Ă̋��ʒ��a�֐�}
  \subsection{�p�^���ʂ̍���}
  \paragraph{�p�^���ʂ̍����̊ȒP�ȗ�}
�X�s���������q�ɂ��čl����B
��1: ��̃X�s��1/2���q�̊��P�b�g�́A�ʒu�ŗL�P�b�g\{$\ket{{\bf x'}}$\}
�Œ������Ԃ�\{$\ket{+}, \ket{-}$\}�Œ�����X�s����ԂƂ̒��ϋ�Ԃ̃P�b�g
$\ket{{\bf x'}, \pm} = \ket{{\bf x'}}\otimes\ket{\pm}$�ł���B

��]�̐������Z�q${\bf J}$�́A
  \begin{displaymath}
{\bf J} = {\bf L}\otimes{\bf 1} + {\bf 1}\otimes{\bf S} = {\bf L} + {\bf S}
  \end{displaymath}

�X�s�������ꗱ�q�̏�ԃP�b�g�́A${\bf L}^{2}, {\bf S}^{2}, L_{z}, S_{z}$�̓����ŗL�P�b�g�œW�J���邱�Ƃ��A
${\bf J}^{2}, J_{z} ,{\bf L}^{2}, {\bf S}^{2}$�̓����ŗL�P�b�g�œW�J���邱�Ƃ��ł���B

��2: �Q�̃X�s��1/2���q�̃X�s�������l����B
�S�̂̃X�s�����Z�q��
  \begin{displaymath}
{\bf S} = {\bf S}_{1}\otimes{\bf 1} + {\bf 1}\otimes{\bf S}_{2} = {\bf S}_{1} + {\bf S}_{2}
  \end{displaymath}

���̌n�̏�ԃP�b�g�́A${\bf S}^{2}, S_{z}$�̓����ŗL�P�b�g$\ket{s,m}$�œW�J���邱�Ƃ��A$S_{1z}, S_{2z}$�̓����ŗL�P�b�g$\ket{m_{1},m_{2}}$�œW�J���邱�Ƃ��ł���B
���̓�ʂ�̕��@�̊��P�b�g�̊Ԃ̊֌W�͈ȉ��̒ʂ�ł���B
  \setcounter{equation}{14}
  \begin{equation}
\ket{1,+1} = \ket{+,+}
  \end{equation}
  \begin{displaymath}
\ket{1,0} = \frac{1}{\sqrt{2}}(\ket{+,-}+\ket{-,+})
  \end{displaymath}
  \begin{displaymath}
\ket{1,-1} = \ket{-,-}
  \end{displaymath}
  \begin{displaymath}
\ket{0,0} = \frac{1}{\sqrt{2}}(\ket{+,-}-\ket{-,+})
  \end{displaymath}
�����̎��̉E�ӂ̌W���̓N���v�V��$\cdot$�S���_���W���̂����Ƃ��ȒP�ȗ�ł���B
  \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$\bracket{j_{1}j_{2};m_{1}m_{2}}{j_{1}j_{2};jm}$�̓N���v�V��$\cdot$�S���_���W���ł���B

�N���v�V��$\cdot$�S���_���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{�N���v�V��$\cdot$�S���_���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{�N���v�V��$\cdot$�S���_���W���ɑ΂���Q����}
  \begin{eqnarray*}
\bra{j_{1}j_{2};m_{1}m_{2}}J_{\pm}/\hbar\ket{j_{1}j_{2};jm}&=&\\
\sqrt{j(j+1)-m(m\pm 1)}\bracket{m_{1},m_{2}}{j,m\pm1}
&=&\\
\sqrt{j_{1}(j_{1}+1)-(m_{1}\mp 1)m_{1}}\bracket{m_{1}\mp 1, m_{2}}{j,m}
&+&\sqrt{j_{2}(j_{2}+1)-(m_{2}\mp 1)m_{2}}\bracket{m_{1},m_{2}\mp 1}{j,m}
  \end{eqnarray*}

note: ��~(\ref{eq1})����B$m_{1}+m_{2}=m\pm 1$�̂Ƃ�0�ɂȂ�Ȃ��B
  \paragraph{�N���v�V��$\cdot$�S���_���W���Ɖ�]�s��}

\end{document}