% <PLAINTEXT>
% $BNL;RJ*M}3X(B $B!'(B $B1i=,LdBj(B 3-5 $B2rEz(B
% $B:n(B  $B@.(B  $B<T(B $B!'(B g040496 $B>BED(B $BOB9,(B
% $B:n@.G/7nF|(B $B!'(B 00/12/06
% $BEE;R%a!<%k(B $B!'(B g040496@mail.ecc.u-tokyo.ac.jp
%
\mondai
{g040496 $B>BED(B $BOB9,(B}
{00/12/06}
{g040496@mail.ecc.u-tokyo.ac.jp}
{3-5}
{$B0lHL$K1i;;;R(B${\cal O}$$B$N0[$J$k8GM-CM(B $E_n$, $E_m$ $B$KBP1~$9$k8GM-4X?t(B $\varphi_n(x), \varphi_m(x)$ $B$OFb@Q$,D>8r$9$k$3$H$r<($;!#(B}
{
$E_n$, $E_m$ $B$O(B ${\cal O}$ $B$N8GM-CM$G$"$k;v$h$j(B
\begin{center}
	$\displaystyle {\cal O}\varphi_n(x) = E_n \varphi_n(x) $

	$\displaystyle {\cal O}\varphi_m(x) = E_m \varphi_n(x) $
\end{center}
$B$3$l$r!"J*M}NL(B${\cal O}$, ${\cal O}^*$ $B$N4|BTCM$r5a$a$k<0!'(B
\begin{center}
	$\displaystyle {\cal O}_{mn} = \int_{-\infty}^{\infty}\varphi_m(x)^* {\cal O} \varphi_n(x) dx $

	$\displaystyle {\cal O}_{nm}^* = \int_{-\infty}^{\infty}\left(\varphi_n(x)^* {\cal O} \varphi_m(x)\right)^* dx $
\end{center}
$B$KBeF~$7$F#2<0$N:9$r$H$j!"(B
\begin{eqnarray}
	{\cal O}_{mn} - {\cal O}_{nm}^* &=& \int_{-\infty}^{\infty}\left\{\left(\varphi_m(x)^* E_n \varphi_n(x) \right) - \left(\varphi_n(x)^* E_m  \varphi_m(x)\right)^* \right\}dx \nonumber\\
	&=& (E_n - E_m)\int_{-\infty}^{\infty} \varphi_m(x)^* \varphi_n(x) dx \nonumber\\
	&=& (E_n - E_m)< \varphi_m | \varphi_n > 
\end{eqnarray}
$B$3$3$G!"0lHL$KJ*M}NL$O4QB,2DG=$J<B?t$J$N$G(B${\cal O}$$B$O%(%k%_!<%H1i;;;R$G$"$k!#8N$K(B
\begin{equation}
	{\cal O}_{mn} = {\cal O}_{nm}^*  \label{eq:3-5-1}
\end{equation}
$B$,@.$jN)$D!#=>$C$F(B $E_m \neq E_n$ $B$h$j(B $m \neq n$ $B$N$H$-(B
\begin{equation}
	< \varphi_m | \varphi_n > = 0 
\end{equation}
$B$,@.N)$9$k!#Kt!"(B $m=n$ $B$N$H$-$OGHF04X?t$N5,3J2=$J$N$GFb@Q$O#1$H$J$k!#(B}
