% <PLAINTEXT>
% $BNL;RJ*M}3X(B $B!'(B $B1i=,LdBj(B 3-6 $B2rEz(B
% $B:n(B  $B@.(B  $B<T(B $B!'(B g040496 $B>BED(B $BOB9,(B
% $B=$!!@5!!<T(B $B!'(B 241025B $B;32<!!??(B
% $B:n@.G/7nF|(B $B!'(B 00/12/06
% $B=$@5G/7nF|(B $B!'(B 02/11/26
% $BEE;R%a!<%k(B $B!'(B g040496@mail.ecc.u-tokyo.ac.jp
% $B=$@5<TEE;R%a!<%k(B $B!'(B g241025@mail.ecc.u-tokyo.ac.jp
%
\mondai
{g040496 $B>BED(B $BOB9,(B}
{01/01/29}
{g040496@mail.ecc.u-tokyo.ac.jp}
{3-6}
{
%----$BLdBjJ8(B----%
$B0J2<$N1i;;;R$N0f8M7?%\%F%s%7%c%k$N8GM-4X?t(B$\varphi_n(x)$(3.30)$B$G$N4|BTCM(B
$B$r5a$a$h!#(B
\begin{center}
	(a) \ \  $x^2$ \ \  (b) \ \  $p^2$ \ \ (c) \ \  $y$ \ ($x$$B$HD>3QJ}8~(B)
\end{center}
}
{
(3.30)$B$O0J2<$NDL$j!#(B
\begin{equation}
	\varphi_n(x) = \cases{
		{\displaystyle \sqrt[]{ \frac{2}{a}}} \cos(k_nx), & (n =
1,3,5,$\cdots$) \cr
		{\displaystyle \sqrt[]{ \frac{2}{a}}} \sin(k_nx), & (n =
2,4,6,$\cdots$) \cr 			}
\end{equation}
$k_n$$B$O(B(3.23)$B$h$j(B\ \ \ $ k_na=n\pi $ .
\begin{equation}
	$B!h(B \ \ \  \sin(k_na)=0, \ \ \ \  \cos(k_na)= (-1)^n
\end{equation}
\paragraph{ (a) }
$x^2$$B$N4|BTCM(B${\cal O} =x^2$$B$O!"(B$n$$B$,4q?t$N$H$-(B
\begin{eqnarray*}
	{\cal O} &=& \int_{-\frac{a}{2}}^{\frac{a}{2}}\varphi_n(x)^* x^2
\varphi_n(x) dx  \\
	&=& \int_{-\frac{a}{2}}^{\frac{a}{2}} {\sqrt[]{\frac{2}{a}}}
\cos(k_nx)^* x^2 {\sqrt[] {\frac{2}{a}}} \cos(k_nx) dx \\
	&=& \frac{2}{a} \int_{-\frac{a}{2}}^{\frac{a}{2}} x^2 \cos^2(k_nx) dx  \\
	&=& \frac{1}{a} \int_{-\frac{a}{2}}^{\frac{a}{2}} x^2 \left\{
1+\cos(2k_nx) \right\} dx \\
	&=& \frac{1}{a} \int_{-\frac{a}{2}}^{\frac{a}{2}}x^2dx +
\underline{\frac{1}{a} \int_{-\frac{a}{2}}^{\frac{a}{2}}x^2\cos(2k_nx)
dx} \\
	&=&
\frac{1}{a}\left[\frac{1}{3}x^3\right]_{-\frac{a}{2}}^{\frac{a}{2}}+\underline{\frac{1}{a}\left\{
\left[
\frac{1}{2k_n}x^2\sin(2k_nx)\right]_{-\frac{a}{2}}^{\frac{a}{2}}-\frac{1}{2k_n}\int_{-\frac{a}{2}}^{\frac{a}{2}}2x\sin(2k_nx)dx\right\}}
\\
	&=& \frac{a^2}{12}-\underline{\frac{1}{k_na}
\int_{-\frac{a}{2}}^{\frac{a}{2}}x\sin(2k_nx)dx} \\
	&=& \frac{a^2}{12}-\underline{\frac{1}{k_na} \left\{ \left[
-\frac{1}{2k_n}x\cos(2k_nx)\right]_{-\frac{a}{2}}^{\frac{a}{2}}+\frac{1}{2k_n}\int_{-\frac{a}{2}}^{\frac{a}{2}}\cos(2k_nx)dx\right\}}
\\
	&=&
\frac{a^2}{12}+\frac{1}{k_n^2a}\left(-\frac{a}{2}\right)-\frac{1}{2k_n^2a}\left[\frac{1}{2k_n}\sin(2k_nx)\right]_{-\frac{a}{2}}^{\frac{a}{2}}\\
	&=& a^2 \left( \frac{1}{12}-\frac{1}{2k_n^2a^2} \right)   \\
	&=& a^2\left(\frac{1}{12}-\frac{1}{2n^2\pi^2}\right)
\end{eqnarray*}
$B$H$J$k!#6v?t$N$H$-$O2<@~It$N7W;;$,4q?t$N$H$-$HA4$/F1$8$G!"(B
\begin{eqnarray*}
	{\cal O} &=& \int_{-\frac{a}{2}}^{\frac{a}{2}}\varphi_n(x)^* x^2
\varphi_n(x) dx  \\
	&=& \int_{-\frac{a}{2}}^{\frac{a}{2}} {\sqrt[]{\frac{2}{a}}}
\sin(k_nx)^* x^2 {\sqrt[] {\frac{2}{a}}} \sin(k_nx) dx \\
	&=& \frac{2}{a} \int_{-\frac{a}{2}}^{\frac{a}{2}} x^2 \sin^2(k_nx) dx  \\
	&=& \frac{1}{a} \int_{-\frac{a}{2}}^{\frac{a}{2}} x^2 \left\{
1-\cos(2k_nx) \right\} dx \\
	&=& \cdots \\
	&=&
\frac{a^2}{12}-\frac{1}{k_n^2a}\left(\frac{a}{2}\right)-\frac{1}{2k_n^2a}\left[\frac{1}{2k_n}\sin(2k_nx)\right]_{-\frac{a}{2}}^{\frac{a}{2}}\\
	&=& a^2 \left( \frac{1}{12}-\frac{1}{2k_n^2a^2} \right)   \\
	&=& a^2\left( \frac{1}{12}-\frac{1}{2n^2\pi^2}\right)
\end{eqnarray*}
$B$h$C$F!"(B$n$$B$N4q6v$K0M$i$:(B$x^2$$B$N4|BTCM$O(B$\displaystyle a^2\left(
\frac{1}{12}-\frac{1}{2n^2\pi^2}\right)$$B$H$J$k!#(B\\
\paragraph{ (b) }
$p^2$$B$N1i;;;R(B${\cal O}$$B$O(B\ \ $\displaystyle {\cal
O}=\left(-i\hbar\frac{d}{dx}\right)^2$$B$H=q$1$F!"(B$n$$B$,4q?t$N$H$-(B
\begin{eqnarray*}
	{\cal O} &=& \int_{-\frac{a}{2}}^{\frac{a}{2}}\varphi_n(x)^* {\cal
O}\varphi_n(x) dx  \\
	&=& \int_{-\frac{a}{2}}^{\frac{a}{2}} {\sqrt[]{\frac{2}{a}}}
\cos(k_nx)^* \left(-i\hbar\frac{d}{dx}\right)^2{\sqrt[] {\frac{2}{a}}}
\cos(k_nx) dx \\
	&=& \frac{2\hbar^2k_n^2}{a} \int_{-\frac{a}{2}}^{\frac{a}{2}}
\cos^2(k_nx) dx  \\
	&=& \frac{\hbar^2k_n^2}{a} \int_{-\frac{a}{2}}^{\frac{a}{2}}  \left\{
1+\cos(2k_nx) \right\} dx \\
	&=& \frac{\hbar^2k_n^2}{a} \left[
x+\frac{1}{2k_n}\sin(2k_nx)\right]_{-\frac{a}{2}}^{\frac{a}{2}} \\
	&=& \hbar^2 k_n^2 \\
	&=& \frac{\hbar^2 n^2\pi^2}{a^2}
\end{eqnarray*}
$B6v?t$N$H$-$bF1MM$K!"(B
\begin{eqnarray*}
	{\cal O} &=& \int_{-\frac{a}{2}}^{\frac{a}{2}} {\sqrt[]{\frac{2}{a}}}
\sin(k_nx)^* \left(-i\hbar\frac{d}{dx}\right)^2{\sqrt[] {\frac{2}{a}}}
\sin(k_nx) dx \\
	&=& \frac{2\hbar^2k_n^2}{a} \int_{-\frac{a}{2}}^{\frac{a}{2}}
\sin^2(k_nx) dx  \\
	&=& \frac{\hbar^2k_n^2}{a} \int_{-\frac{a}{2}}^{\frac{a}{2}}  \left\{
1-\cos(2k_nx) \right\} dx \\
	&=& \frac{\hbar^2k_n^2}{a} \left[
x-\frac{1}{2k_n}\sin(2k_nx)\right]_{-\frac{a}{2}}^{\frac{a}{2}} \\
	&=& \frac{\hbar^2 n^2\pi^2}{a^2}
\end{eqnarray*}
$B$h$C$F!"(B$n$$B$N4q6v$K0M$i$:(B$p^2$$B$N4|BTCM$O(B$\displaystyle \frac{\hbar^2
n^2\pi^2}{a^2}$$B$H$J$k!#(B\\
\paragraph{ (c) }
$y$$B$O(B$x$$B$HD>8r$7$F$$$k$N$G!"(B$x$$B$G@QJ,$9$k>l9g$K$ODj?t$H8+$J$;$k!#(B\\
$\varphi_n(x)$$B$O4{$K5,3J2=$5$l$F$$$k$N$G!"(B$y$$B$N4|BTCM$r(B${\cal O}$$B$H$7$F(B
\begin{eqnarray*}
	{\cal O} &=& \int_{-\frac{a}{2}}^{\frac{a}{2}}\varphi_n(x)^* y
\varphi_n(x) dx  \\
	&=& y\int_{-\frac{a}{2}}^{\frac{a}{2}}\varphi_n(x)^* \varphi_n(x) dx  \\
	&=& y*1 \\
	&=& y
\end{eqnarray*}
$B$G$"$k!#$h$C$F(B$y$$B$N4|BTCM$O(B$y$$B!#(B
}

