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{g240916 $B9b>k=E9((B}
{/02/10/10}
{g240916@mail.ecc.u-tokyo.ac.jp}
{3-7}
{%----$BLdBjJ8(B----%
$(3.30)$$B$K<($7$?0f8M7?%]%F%s%7%c%k$N8GM-4X?t(B$\varphi_n(x)$$B$KBP$7!"(B${\cal O}$$B$N9TNs(B$<\varphi_{n} |{\cal O}| \varphi_{n+1}>$$B$r7W;;$;$h!#(B
\begin{center}
	(a) \ \  $x$ \ \  (b) \ \  $x^2$ \ \ (c) \ \  $\frac{d}{dx}$ \ 
\end{center}
}
{%----$B2rEzJ8(B----%
\begin{description}
\item[(a)]
${\cal O}=x$B$N$H$-(B$
\[<\varphi_{n}|{\cal O}|\varphi_{n+1}>=\int_{-\frac{a}{2}}^{\frac{a}{2}}\varphi_{n}(x)x\varphi_{n+1}(x)dx\]\\
\begin{enumerate}
\item $n$$B$,6v?t$N$H$-(B\\
\begin{eqnarray}
<\varphi_{n}|{\cal O}|\varphi_{n+1}>
&=&\int_{-\frac{a}{2}}^{\frac{a}{2}} \sqrt{\frac{2}{a}}\sin{(k_{n}x)}x\sqrt{\frac{2}{a}}\cos{(k_{n+1}x)}dx\nonumber\\
&=&\frac{2}{a}\int_{-\frac{a}{2}}^{\frac{a}{2}}\sin{(k_{n}x)}x\cos{(k_{n+1}x)}dx\nonumber\\
&=&\frac{1}{a}\int_{-\frac{a}{2}}^{\frac{a}{2}}x\left(\sin{(k_{n}+k_{n+1})x}+\sin{(k_{n}-k_{n+1})x}\right)dx\nonumber\\
&=&\frac{1}{a}\int_{-\frac{a}{2}}^{\frac{a}{2}}x\left(\sin{\frac{(2n+1)\pi}{a}x}+\sin{(-\frac{\pi}{a})x}\right)dx\\
&(&k_{n}=\frac{n\pi}{a}$B$rBeF~$7$?(B)\nonumber
\end{eqnarray}
$B$3$3$G(B
\begin{eqnarray}
\int{x\sin{qx}dx}&=&x(-\frac{1}{q}\cos{qx})+\frac{1}{q}\int\cos{qx}dx+C\nonumber\\
&=&-\frac{x}{q}\cos{qx}+\frac{1}{q^2}\sin{qx}+C^{\prime}\nonumber
\end{eqnarray}
$B$G$"$k$+$i(B
\begin{eqnarray}
<\varphi_{n}|{\cal O}|\varphi_{n+1}>&=&\frac{2a}{\pi^2}\{\frac{1}{(2n+1)^2}\sin{(n+\frac{1}{2})\pi}-1\}\nonumber\\
&=&\frac{2a}{\pi^2}\{\frac{1}{(2n+1)^2}-1\}\nonumber\\
&=&-\frac{8n(n+1)}{(2n+1)^2\pi^2}a
\end{eqnarray}
\item $n$$B$,4q?t$N$H$-(B\\
\end{enumerate}
\begin{eqnarray}
<\varphi_{n}|{\cal O}|\varphi_{n+1}>
&=&\int_{-\frac{a}{2}}^{\frac{a}{2}}\sqrt{\frac{2}{a}}\cos{(k_{n}x)}x\sqrt{\frac{2}{a}}\sin{(k_{n+1}x)}dx\nonumber\\
&=&\frac{1}{a}\int_{-\frac{a}{2}}^{\frac{a}{2}}x\left(\sin{\frac{(2n+1)\pi}{a}x}+\sin{(\frac{\pi}{a})x}\right)dx\\
&=&\frac{2a}{\pi^2}\{-\frac{1}{(2n+1)^2}\sin{(n+\frac{1}{2})\pi}+1\}\nonumber\\
&=&\frac{8n(n+1)}{(2n+1)^2\pi^2}a
\end{eqnarray}
$B0J>e(B(2)(4)$B$r$R$H$D$N<0$K$7$F(B\\
\[<\varphi_{n}|{\cal O}|\varphi_{n+1}>=\frac{(-1)^{n+1}8n(n+1)}{(2n+1)^2\pi^2}a\]\\
\item[(b)]
${\cal O}=x^2$B$N$H$-(B$
\[<\varphi_{n}|{\cal O}|\varphi_{n+1}>=\int_{-\frac{a}{2}}^{\frac{a}{2}}\varphi_{n}(x)x^2\varphi_{n+1}(x)dx\]\\
$(a)$(1)(3)$B$^$G$NF1MM$N7W;;$K$h$j(B\\
\begin{enumerate}
\item $n$$B$,6v?t$N$H$-(B\\
\[<\varphi_{n}|{\cal O}|\varphi_{n+1}>=\frac{1}{a}\int_{-\frac{a}{2}}^{\frac{a}{2}}x^2\{\sin{\frac{(2n+1)\pi}{a}x}+\sin{(-\frac{\pi}{a})x}\}dx\]\\
\item $n$$B$,4q?t$N$H$-(B\\
\[<\varphi_{n}|{\cal O}|\varphi_{n+1}>=\frac{1}{a}\int_{-\frac{a}{2}}^{\frac{a}{2}}x^2\{\sin{\frac{(2n+1)\pi}{a}x}+\sin{\frac{\pi}{a}x}\}dx\]\\
\end{enumerate}
$1,2$B$H$b$K4q4X?t$N@QJ,$G$"$k$?$a(B$\\
\[<\varphi_{n}|{\cal O}|\varphi_{n+1}>=0\]\\
\\
\item[(c)]
${\cal O}=\displaystyle{\frac{d}{dx}}$B$N$H$-(B$
\begin{enumerate}
\item $n$$B$,6v?t$N$H$-(B\\
\begin{eqnarray}
<\varphi_{n}|{\cal O}|\varphi_{n+1}>
&=&\int_{-\frac{a}{2}}^{\frac{a}{2}} \sqrt{\frac{2}{a}}\sin{(k_{n}x)}\frac{d}{dx}\sqrt{\frac{2}{a}}\cos{(k_{n+1}x)}dx\nonumber\\
&=&\frac{2}{a}\int_{-\frac{a}{2}}^{\frac{a}{2}}\sin{(k_{n}x)}(-k_{n+1}\sin{(k_{n+1}x)})dx\nonumber\\
&=&-\frac{1}{a}k_{n+1}\int_{-\frac{a}{2}}^{\frac{a}{2}}{\cos(k_n-k_{n+1})x-\cos(k_n+k_{n+1})x}dx\nonumber\\
&=&-\frac{2}{a}k_{n+1}\left(\frac{1}{k_n-k_{n+1}}\sin{(k_n-k_{n+1})}\frac{a}{2}-\frac{1}{k_n+k_{n+1}}\sin{(k_n+k_{n+1})}\frac{a}{2}\right)\nonumber
\end{eqnarray}
$B$h$C$F(B$k_{n}=\displaystyle{\frac{n\pi}{a}}$$B$rBeF~$7$F(B\\
\begin{eqnarray}
<\varphi_{n}|{\cal O}|\varphi_{n+1}>=-\frac{4n(n+1)}{(2n+1)a}
\end{eqnarray}
\item $n$$B$,4q?t$N$H$-(B\\
\begin{eqnarray}
<\varphi_{n}|{\cal O}|\varphi_{n+1}>
&=&\int_{-\frac{a}{2}}^{\frac{a}{2}} \sqrt{\frac{2}{a}}\cos{(k_{n}x)}\frac{d}{dx}\sqrt{\frac{2}{a}}\sin{(k_{n+1}x)}dx\nonumber\\
&=&\frac{2}{a}\int_{-\frac{a}{2}}^{\frac{a}{2}}\cos{(k_{n}x)}(k_{n+1}\cos{(k_{n+1}x)})dx\nonumber\\
&=&\frac{1}{a}k_{n+1}\int_{-\frac{a}{2}}^{\frac{a}{2}}{\cos(k_n+k_{n+1})x+\cos(k_n-k_{n+1})x}dx\nonumber\\
&=&\frac{2}{a}k_{n+1}\left(\frac{1}{k_n+k_{n+1}}\sin{(k_n+k_{n+1})}\frac{a}{2}+\frac{1}{k_n-k_{n+1}}\sin{(k_n-k_{n+1})}\frac{a}{2}\right)\nonumber
\end{eqnarray}
$B$h$C$F(B$k_{n}=\displaystyle{\frac{n\pi}{a}}$$B$rBeF~$7$F(B\\
\begin{eqnarray}<\varphi_{n}|{\cal O}|\varphi_{n+1}>=\frac{4n(n+1)}{(2n+1)a}
\end{eqnarray}
\end{enumerate}
$B0J>e(B(5)(6)$B$r$R$H$D$N<0$K$7$F(B\\
\[<\varphi_{n}|{\cal O}|\varphi_{n+1}>=\frac{(-1)^{n+1}4n(n+1)}{(2n+1)a}\]
\end{description}
}


