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% $B:n(B  $B@.(B  $B<T(B : g940455 $BIMCO?50lO:(B
% $B:n@.G/7nF|(B : /99/11/30
% $BEE;R%a!<%k(B : g940455@mail.ecc.u-tokyo.ac.jp
%
% \mondai{$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
{g940455 $BIMCO?50lO:(B}
{99/11/30}
{g940455@mail.ecc.u-tokyo.ac.jp}
{3-3}
{$B0f8M7?%]%F%s%7%c%k(B (3.15) $B$NCM$,(B $0$ $B$K$J$k(B $x$ $B$NHO0O$r(B $|x|<a/2$ $B$+$i!"(B$0<x<a$ $B$K$7$?$H$-$N8GM-CM!"8GM-4X?t$r5a$a$h!#(B}
{
$B$^$:(B$x>a$$B$^$?$O(B$x<0$$B$NItJ,$G$O!"%]%F%s%7%c%k$,L58BBg$J$N$GEE;R$OB8:_$7$J$$$H9M$($FNI$$!#<!$K(B $0\leq{x}\leq{a}$ $B$NItJ,$K$D$$$F!"(B$U(x)=0$$B$h$j%O%_%k%H%K%"%s$O!"(B
\begin{equation}
{\cal H}=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}
\end{equation}
$B$G$"$k$+$i!"%7%e%l!<%G%#%s%,!<J}Dx<0$O!"(B
\begin{equation}
\left(-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}-E\right)\varphi (x)=0\label{eq:Q3-3-1}
\end{equation}
$B$H$J$j!"$3$3$G!"(B
\begin{equation}
k=\sqrt{\frac{2mE}{\hbar^2}} \label{eq:Q3-3-2}
\end{equation}
$B$H$*$/$H!"(B(\ref{eq:Q3-3-1})$B$O!"(B
\begin{equation}
\left(\frac{d^2}{dx^2}+k^2\right)\varphi (x)=0 \label{eq:Q3-3-3}
\end{equation}
$B$H$J$k!#$3$N<0$N0lHL2r$O!"(B
\begin{equation}
\varphi (x)=C_1e^{ikx}+C_2e^{-ikx}, \: (C_1,C_2\mbox{$B$OG$0UDj?t(B})
\label{eq:Q3-3-4}
\end{equation}
$B$G$"$k!#$3$3$G6-3&>r7o!"(B
\begin{equation}
\varphi (0)=0,\: \varphi (a)=0
\end{equation}
$B$h$j!"(B
\begin{equation}
\left(
\begin{array}{cc}
1 & 1 \\
e^{ika} & e^{-ika}
\end{array}
\right)
\left(
\begin{array}{c}
C_1 \\
C_2
\end{array}
\right)
=
\left(
\begin{array}{c}
0 \\
0
\end{array}
\right)
\label{eq:Q3-3-5}
\end{equation}
$B$H$$$&O"N)J}Dx<0$,F@$i$l$k!#$3$NO"N)J}Dx<0$,(B $C_1=C_2=0$ $B0J30$N2r$r;}$D$?$a$K$O!"(B
\begin{equation}
\left|
\begin{array}{cc}
1 & 1 \\
e^{ika} & e^{-ika}
\end{array}
\right|=e^{-ika}-e^{ika}=-2i\sin(ka)=0
\end{equation}
$B$G$J$1$l$P$$$1$J$$!#$7$?$,$C$F!"GH?t(B $k$ $B$O!"(B
\begin{equation}
k_na=n\pi, \: (n=1,2,\cdots ) \label{eq:Q3-3-6}
\end{equation}
$B$H$J$k!#$3$N(B $k_n$ $B$NCM$r(B(\ref{eq:Q3-3-2}$B$KBeF~$9$k$H!"%(%M%k%.!<8GM-CM(B $E_n$ $B$H$7$F!"(B
\begin{equation}
E_n=\frac{\hbar^2}{2m}\frac{\pi^2}{a^2}n^2, \: (n=1,2,\cdots )
\end{equation}
$B$rF@$k!#<!$K!"(B(\ref{eq:Q3-3-6}) $B$r(B (\ref{eq:Q3-3-5}) $B$KBeF~$9$k$H!"(B
\begin{equation}
C_1+C_2=0, \: \mbox{$B$9$J$o$A(B} C_1:C_2=1:-1
\end{equation}
$B$G$"$k;v$,$o$+$k!#$3$N4X78$r(B (\ref{eq:Q3-3-4}) $B$KE,MQ$9$k$H!"(B
\begin{equation}
\varphi_n(x)=C'\{e^{ik_nx}-e^{-ik_nx}\} \\
=C\sin(k_nx)
\end{equation}
$B$H$$$&8GM-4X?t$,5a$^$k!#$3$3$G5,3J2=>r7o!"(B
\begin{equation}
\int_{-\infty}^{\infty}|\varphi_n(x)|^2dx=1
\end{equation}
$B$r9MN8$KF~$l$F5,3J2=$r9T$&$H!"(B
\begin{equation}
\int_{-\infty}^{\infty}|\varphi_n(x)|^2dx=C^2\int_0^a\sin^2(k_nx)dx=\frac{C^2a}{2}=1
\end{equation}
$B$G$"$k$+$i!"(B$C=\sqrt{2/a}$ $B$G$"$k!#$h$C$F!"5,3J2=$5$l$?GHF04X?t$O!"(B
\begin{equation}
\varphi_n(x)=\sqrt{\frac{2}{a}}\sin(k_nx)
\end{equation}
$B$H$J$k!#(B
}


