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% $B=$(B  $B@5(B  $B<T(B : g240916 $B9b>k(B  $B=E9((B
% $B:n@.G/7nF|(B : /99/11/
% $B=$@5G/7nF|(B : /02/10/16
% $BEE;R%a!<%k(B : g941000@mail.ecc.u-tokyo.ac.jp
% $B=$@5<TEE;R%a!<%k(B : g240916@mail.ecc.u-tokyo.ac.jp
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% \mondai{$B:n@.<T(B}{$B:n@.G/7nF|(B}{$BEE;R%a!<%k(B}{$BLdBjHV9f(B}{$BLdBjJ8(B}{$B2rEz(B}
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\mondai
{941000B $B8eF#(B  $BGn;V(B}
{99/11/8}
{g941000@mail.ecc.u-tokyo.ac.jp}
{3-4}
{$B0f8M7?%]%F%s%7%c%k$N8GM-4X?t(B$\phi_n(x)$(3.30)$B$GFb@Q(B$<\phi_n|\phi_m>$
$B$,0J2<$N4X78<0$rK~$?$9$3$H$r<($;!#(B
\begin{center}
$\displaystyle <\varphi_n|\varphi_m>=\int_{-a/2}^{a/2} \varphi_n^{\ast} (x)
 \varphi_m(x) dx
 =\delta_{mn}$
\end{center}
$B$3$3$G(B$\delta_{mn}$$B$O!"(B$n$$B$H(B$m$$B$,F1$8>l9g#1$G!"0[$J$k>l9g#0$rM?$($k4X?t(B
\begin{center}
$\delta_{mn}= \cases{
1 & ($n = m$) \cr
0 & ($n \neq m$) \cr}$
\end{center}
$B$G%/%m%M%C%+!<$N%G%k%?$H8F$V!#(B}
{
(3.30)$B$h$j!"(B
\begin{equation}
\displaystyle \varphi_n = \left\{
\begin{array}{ll}
\sqrt{\frac{2}{a}} \sin k_nx  & (n=2,4,6,\cdots \cdots)\\
\sqrt{\frac{2}{a}} \cos k_nx  & (n=1,3,5,\cdots \cdots)
\end{array} \right.
\end{equation}
\begin{equation}
\displaystyle k_n = \frac{n\pi}{a}
\end{equation}
$B$G$"$k!#$3$3$G(B
(i)(a)$n = m$$B$G(B$n$$B$,6v?t$N$H$-!"(B
\begin{eqnarray}
\displaystyle($B:8JU(B)&=&\frac{2}{a} \int_{-a/2}^{a/2}  \sin^2 k_nx  dx\nonumber\\
\displaystyle&=&\frac{2}{a} \int_{-a/2}^{a/2}\frac{1-\cos 2k_nx}{2}  
dx\nonumber\\
\displaystyle&=&
\frac{1}{a}\left[x-\frac{\sin 2k_nx}{2k_n} \right]_{-a/2}^{a/2}\nonumber\\
&=&\frac{1}{2}-\frac{\sin ak_n}{2ak_n}+\frac{1}{2}
+\frac{\sin (-ak_n)}{2ak_n}\nonumber\\
&=&1-\frac{\sin n\pi}{n\pi}\nonumber\\
&=& 1 
\end{eqnarray}
(b)$n = m$$B$G(B$n$$B$,4q?t$N$H$-!"(B
\begin{eqnarray}
\displaystyle($B:8JU(B)&=&\frac{2}{a}\int_{-a/2}^{a/2}\cos^2 k_nx  dx\nonumber\\
\displaystyle&=&\frac{2}{a} \int_{-a/2}^{a/2}\frac{1+\cos 2k_nx}{2}
dx\nonumber\\
\displaystyle&=&\frac{1}{a} \left[x+\frac{\sin 2k_nx}{2k_n} \right]
_{-a/2}^{a/2}\nonumber\\
&=&\frac{1}{2}+\frac{\sin ak_n}{2ak_n} +
 \frac{1}{2}-\frac{\sin (-ak_n)}{2ak_n}\nonumber\\
&=&1
\end{eqnarray}
$B$^$?!"(B(ii)(a)$n \neq  m$ $B$G(B$n$B!"(Bm$$B$H$b$K6v?t$N$H$-!"(B
\begin{eqnarray}
\displaystyle ($B:8JU(B) 
&=&\frac{2}{a}\int_{-a/2}^{a/2}\sin k_nx \sin k_mx dx\nonumber\\
\displaystyle &=&-\frac{1}{a} \int_{-a/2}^{a/2} \{ \cos(k_n+k_m)x 
- \cos(k_n-k_m)x \} dx\nonumber\\
\displaystyle &=&-\frac{1}{a}\int_{-a/2}^{a/2} \{ \cos \frac{(n+m)\pi}{a}x 
- \cos \frac{(n-m)\pi}{a}x \} dx\nonumber\\
\displaystyle &=&-\frac{1}{a} \left[  \frac{a}{(n+m)\pi}
\sin \frac{(n+m)\pi}{a}x
- \frac{a}{(n-m)\pi} \sin \frac{(n-m)\pi}{a}x \right]_{-a/2}^{a/2}\nonumber\\ 
\displaystyle&=& -\frac{2}{a} \{ \frac{a}{(n+m)\pi}\sin \frac{(n+m)\pi}{2}
- \frac{a}{(n-m)\pi} \sin \frac{(n-m)\pi}{2}\}
\end{eqnarray}
$B$3$3$G!"(B$n$B!"(Bm$$B$,6v?t$h$j!"(B$n+m$B!"(Bn-m$$B$b6v?t$G$"$k$+$i(B
$\frac{(n+m)\pi}{2}$B!"(B\frac{(n-m)\pi}{2}$$B$O(B$\pi$$B$N@0?tG\$H$J$j!"(B
$\sin \frac{(n+m)\pi}{2}=\sin \frac{(n-m)\pi}{2}=0$$B$h$j(B
$B:8JU$O#0$G$"$k!#(B\\
(b)$n \neq  m$ $B$G(B$n$B!"(Bm$$B$H$b$K4q?t$N$H$-!"(B
\begin{eqnarray}
\displaystyle ($B:8JU(B) 
&=& \frac{2}{a}\int_{-a/2}^{a/2}\cos k_nx \cos k_mx dx\nonumber\\
\displaystyle&=&\frac{1}{a}\int_{-a/2}^{a/2} \{ \cos(k_n+k_m)x 
+ \cos(k_n-k_m)x \} dx\nonumber\\
\displaystyle &=&\frac{1}{a}\int_{-a/2}^{a/2} \{ \cos \frac{(n+m)\pi}{a}x 
+ \cos \frac{(n-m)\pi}{a}x \} dx\nonumber\\
\displaystyle &=&\frac{1}{a} \left[  \frac{a}{(n+m)\pi}
\sin \frac{(n+m)\pi}{a}x
+ \frac{a}{(n-m)\pi} \sin \frac{(n-m)\pi}{a}x \right]_{-a/2}^{a/2}\nonumber\\ 
\displaystyle &=&\frac{2}{a} \{ \frac{a}{(n+m)\pi}\sin \frac{(n+m)\pi}{2}
+ \frac{a}{(n-m)\pi} \sin \frac{(n-m)\pi}{2} \}
\end{eqnarray}
$B$3$3$G!"(B$n$B!"(Bm$$B$,4q?t$h$j!"(B$n+m$B!"(Bn-m$$B$O6v?t$G$"$k$+$i(B
$\frac{(n+m)\pi}{2}$B!"(B\frac{(n-m)\pi}{2}$$B$O(B$\pi$$B$N@0?tG\$H$J$j!"(B
$\sin \frac{(n+m)\pi}{2}=\sin \frac{(n-m)\pi}{2}=0$$B$h$j(B
$B:8JU$O#0$G$"$k!#(B\\
(c)$n \neq  m$ $B$G(B$n$B!"(Bm$$B$N6v4q$,0lCW$7$J$$$H$-!":8JU$O(B
$\cos$$B$H(B$\sin$$B$N@Q$N@QJ,$H$J$k!#6v4X?t$H4q4X?t$N@Q$h$j!":8JU$O(B
$B4q4X?t$N@QJ,$H$J$k!#@QJ,6h4V$,86E@$K$D$$$FBP>N$G$"$k$+$i!":8JU(B
$B$N@QJ,$r<B9T$9$k$H#0$rF@$k!#(B\\
$B$h$C$F(B (i)(ii)$B$h$jBj0U$O<($5$l$?!#(B}



