\documentclass{jarticle}
\title{Note.(2)}
\author{ʸ��: ���� ��Ϻ}
\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{�ѱ�ư�̤θ�ͭ�ͤȸ�ͭ����}
  \paragraph{�򴹴ط��ȤϤ����黻��}
  \begin{equation}
{\bf J}^{2} \equiv J_{x}^{2} + J_{y}^{2} + J_{z}^{2}
  \end{equation}
����������${\bf J}^{2}$��$J_{k}$�ȸ򴹤���:
  \begin{equation}
[{\bf J}^{2}, J_{k}] = 0
  \end{equation}

${\bf J}^{2}$��$J_{z}$���򴹤���Ȥ���ȡ�������Ʊ����ͭ���å�$\ket{a,b}$�����äƼ���������Ω�ġ�
  \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}

�����ǡ��Ϥ����黻��
  \begin{equation}
J_{\pm} \equiv  J_{x} \pm i J_{y}
  \end{equation}
���������ȡ�$J_{\pm}$��$\ket{a,b}$�˼��Τ褦�˺��Ѥ���:
  \setcounter{equation}{11}
  \begin{equation}
J_{\pm}\ket{a,b} = c_{\pm}\ket{a, b \pm \hbar}
  \end{equation}

$c_{\pm}$�ˤĤ��ƤϹ������Ǥ����Ȥ��˵��롣

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{${\bf J}^{2}$�����$J_{z}$�θ�ͭ��}
  \begin{equation}
a \geq b^{2}
  \end{equation}
�Ǥ��뤳�Ȥ���${\bf J}^{2}-J_{z}^{2}$�δ����ͤ�ͤ��뤳�ȤǤ狼�롣

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

�����ꡢb�ˤϾ岼�¤�¸�ߤ��뤳�Ȥ��狼��Τǡ�������������ɬ�פ����롣
  \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}

�����ǡ�
  \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}
�Ǥ��뤫�顢
  \begin{displaymath}
a = b_{max}(b_{max} + \hbar) =  (-b_{min})((-b_{mix}) + \hbar)
  \end{displaymath}
��äơ�
  \setcounter{equation}{25}
  \begin{equation}
b_{max} = - b_{min}
  \end{equation}
�������롣

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

�����ꡢ$\ket{a, -b_{max}}$��$J_{+}$��n����Ѥ������$\ket{a, b_{max}}$�ˤʤ�Ȥ���Ȥ���
  \setcounter{equation}{28}
  \begin{equation}
b_{max} = \frac{n\hbar}{2}
  \end{equation}
�Ǥ��롣�����ǡ�
  \begin{equation}
j \equiv \frac{b_{max}}{\hbar}
  \end{equation}
  \setcounter{equation}{31}
  \begin{equation}
b \equiv m\hbar
  \end{equation}
���������ȡ�
  \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}
�Ȥʤ롣�椨�ˡ�
${\bf J}^{2}$��$J_{z}$��Ʊ����ͭ���åȤ�$\ket{j,m}$�Ǥ���魯�Ȥ���ͭ����������
  \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}
�Ȥʤ롣
  \paragraph{�ѱ�ư�̱黻�Ҥι�������}
  \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: $J_{\pm}\ket{j,m} = c_{\pm}\ket{j,m\pm 1}, J_{\mp}J_{\pm} = {\bf J}^{2}-J_{z}^{2}\mp\hbar J_{z}, \mbox{�Υ��} = |c_{\pm}|^{2} = \{j(j+1)-m(m\pm 1)\}\hbar^{2}$
  \paragraph{��ž�黻�Ҥ�ɽ��}
��ž�黻��${\cal D}(R)$�ι������Ǥϡ���ž$R$��${\bf n}$��$\phi$�ǻ��ꤵ���Ȥ���
  \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{��ƻ�ѱ�ư��}
  \setcounter{equation}{0}
��ƻ�ѱ�ư��${\bf L}$��
  \begin{equation}
{\bf L} \equiv {\bf x} \times {\bf p}
  \end{equation}
��������롣��γ�ҤˤĤ���������ͳ�٤�̵��Ǥ������${\bf L}$��${\bf J}$�˰��פ��뤳�Ȥ�ʲ��Ǽ�����
  \paragraph{��ž�������黻�ҤȤ��Ƥε�ƻ�ѱ�ư��}
${\bf L}$�ϡ�
  \begin{itemize}
    \item{�ѱ�ư�̤θ򴹴ط� $[L_{x}, L_{y}] = i\hbar L_{z}$}
    \item{��ž������ $({\bf 1} - iL_{z}\delta\phi/\hbar)\ket{x',y',z'} = \ket{x'-y'\delta\phi, y'+x'\delta\phi, z'}$}
  \end{itemize}
����­���롣

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

$\bracket{x',y',z'}{\alpha}$��Ϳ��������ư�ؿ�����İ�γ�Ҥ�z���ޤ���̵�¾���ž�����뤳�Ȥ�ͤ��롣
  \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*}
��ꡢ
  \setcounter{equation}{8}
  \begin{equation}
\bra{{\bf x'}}L_{z}\ket{\alpha} = -i\hbar\frac{\partial}{\partial\phi}\bracket{{\bf x'}}{\alpha}
  \end{equation}
�������롣

$L_{x}, L_{y}, L_{\pm}$�ˤĤ��Ƥ�Ʊ�ͤθ�Ƥ��Ԥ����Ȥˤ�ꡢ
  \begin{displaymath}
\bra{{\bf x'}}{\bf L}^{2}\ket{\alpha} = -\hbar^{2}\Lambda^{2}\bracket{{\bf x'}}{\alpha}
  \end{displaymath}
�Ȥ����롣

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{����Ĵ�´ؿ�}
���оΤΥݥƥ󥷥��Τ�Ȥǥ��ԥ������ʤ���Ĥ�γ�Ҥ�ͤ��롣
���ͥ륮����ͭ�ؿ���
  \setcounter{equation}{21}
  \begin{equation}
\bracket{{\bf x'}}{n, l, m} = R_{nl}(r)Y_{l}^{m}(\theta, \phi)
  \end{equation}

note: $H$�����оΤʤȤ�$H$��$L_{z}, {\bf L}^{2}$�ȸ򴹡����ͥ륮����ͭ���åȤ⵰ƻ�ѱ�ư�̤θ�ͭ���åȡ�

���ٰ�¸���ˤĤ��ƹͤ��롣
������ͭ���å�$\ket{{\bf n}}$������������ٰ�¸��ʬ��ʬΥ�������ε���Ĵ�´ؿ���ͤ��롣
  \begin{equation}
\bracket{{\bf n}}{l,m} = Y_{l}^{m}(\theta, \phi) = Y_{l}^{m}({\bf n})
  \end{equation}

��ƻ�ѱ�ư�̤θ�ͭ���å�$\ket{l,m}$��ޤ�ط��������б��������Ĵ�´ؿ���ޤ�ط����������롣

��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}
�������롣$Y_{l}^{m}(\theta, \phi)$��$\phi$��¸��ʬ��$e^{im\phi}$�η��ˤʤ뤳�Ȥ��狼�롣

��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}
�������롣

note: �ޤ�������$\bra{{\bf n}}$�򤫤��롣

��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}
�������롣

note: �ޤ��������$\int d\Omega_{{\bf n}}\ket{{\bf n}}\bra{{\bf n}} = 1$���������롣

  \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{��ž����Ȥ��Ƥε���Ĵ�´ؿ�}
  \subsection{�ѱ�ư�̤ι���}
  \paragraph{�ѱ�ư�̤ι����δ�ñ����}
���ԥ�����γ�ҤˤĤ��ƹͤ��롣
��1: ��ĤΥ��ԥ�1/2γ�Ҥδ��쥱�åȤϡ����ָ�ͭ���å�\{$\ket{{\bf x'}}$\}
��ĥ������֤�\{$\ket{+}, \ket{-}$\}��ĥ���륹�ԥ���֤Ȥ�ľ�Ѷ��֤Υ��å�
$\ket{{\bf x'}, \pm} = \ket{{\bf x'}}\otimes\ket{\pm}$�Ǥ��롣

��ž�������黻��${\bf J}$�ϡ�
  \begin{displaymath}
{\bf J} = {\bf L}\otimes{\bf 1} + {\bf 1}\otimes{\bf S} = {\bf L} + {\bf S}
  \end{displaymath}

���ԥ����İ�γ�Ҥξ��֥��åȤϡ�${\bf L}^{2}, {\bf S}^{2}, L_{z}, S_{z}$��Ʊ����ͭ���åȤ�Ÿ�����뤳�Ȥ⡢
${\bf J}^{2}, J_{z} ,{\bf L}^{2}, {\bf S}^{2}$��Ʊ����ͭ���åȤ�Ÿ�����뤳�Ȥ�Ǥ��롣

��2: ���ĤΥ��ԥ�1/2γ�ҤΥ��ԥ�����ͤ��롣
���ΤΥ��ԥ�黻�Ҥ�
  \begin{displaymath}
{\bf S} = {\bf S}_{1}\otimes{\bf 1} + {\bf 1}\otimes{\bf S}_{2} = {\bf S}_{1} + {\bf S}_{2}
  \end{displaymath}

���ηϤξ��֥��åȤϡ�${\bf S}^{2}, S_{z}$��Ʊ����ͭ���å�$\ket{s,m}$��Ÿ�����뤳�Ȥ⡢$S_{1z}, S_{2z}$��Ʊ����ͭ���å�$\ket{m_{1},m_{2}}$��Ÿ�����뤳�Ȥ�Ǥ��롣
�������̤����ˡ�δ��쥱�åȤδ֤δط��ϰʲ����̤�Ǥ��롣
  \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}
�����μ��α��դη����ϥ���ץ���$\cdot$������󷸿��Τ�äȤ��ñ����Ǥ��롣
  \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����������$\bracket{j_{1}j_{2};m_{1}m_{2}}{j_{1}j_{2};jm}$�ϥ���ץ���$\cdot$������󷸿��Ǥ��롣

����ץ���$\cdot$������󷸿�������:
  \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{����ץ���$\cdot$������󷸿��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{����ץ���$\cdot$������󷸿����Ф���������}
  \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})���顣$m_{1}+m_{2}=m\pm 1$�ΤȤ�0�ˤʤ�ʤ���
  \paragraph{����ץ���$\cdot$������󷸿��Ȳ�ž����}

\end{document}