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% $B:n(B  $B@.(B  $B<T(B $B!'(B 240916J $B9b>k(B $B=E9((B
% $B=$!!@5!!<T(B $B!'(B 240509f $B;-Eg(B $B>;90(B
% $B:n@.G/7nF|(B $B!'(B /02/10/10
% $B=$@5G/7nF|(B $B!'(B /02/12/02
% $BEE;R%a!<%k(B $B!'(B g240916@mail.ecc.u-tokyo.ac.jp
% $B=$@5<TEE;R%a!<%k(B $B!'(B g240509@mail.ecc.u-tokyo.ac.jp
%
% \mondai{$B:n@.<T(B}{$B:n@.G/7nF|(B}{$BEE;R%a!<%k(B}{$BLdBjHV9f(B}{$BLdBjJ8(B}{$B2rEz(B}
%
\mondai
{240916J $B9b>k(B $B=E9((B}
{/02/10/10,/02/12/02 modified}
{g240916@mail.ecc.u-tokyo.ac.jp}
{4-6}
{
%----$BLdBjJ8(B----%
$B<!$N%]%F%s%7%c%k%(%M%k%.!<(B$ U(x) $ ($B?^#3;2>H!K(B
\[ U(x) = \left\{
\begin{array}{ll}
 \infty     &(x<-b)\\
 -V_{0}  & (-b<x < -a) \\
0&(-a<x<a)\\
 -V_{0} & (a < x < b) \\
 \infty     & (b < x)
\end{array} \right. \]
$B$N8GM-CM$r5a$a$k<0$N7A$r<($;!#(B\\
}
{
%----$B2rEzJ8(B---%

\begin{enumerate}
\item$-b<x<-a$$B$N$H$-(B
$B?^#3$N%0%i%U$r(B$x$$BJ}8~$K(Bb$BJ?9T0\F0$5$;$k$H!J(B$x\rightarrow x-b$$B!K!"(B$U(x)=-V_0$B$H$J$k(Bx$B$NHO0O$O(B
0<x<b-a$$B$H$J$k!#$3$N$H$-%7%e%l%G%#%s%,!<J}Dx<0$O(B
\begin{equation}
\left\{-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}-(E+V_0)\right\}\varphi(x)=0
\end{equation}
$B$h$C$F(B
\begin{equation}
 \left( \frac{d^{2}}{dx^{2}} + k^{2} \right) \varphi(x) = 0 
\end{equation}
\begin{equation}
 k=\sqrt{\frac{2m(V_{0}+E)}{\hbar^{2}}}
\end{equation}
$B$3$l$r2r$/$H!"(B
\begin{equation}
 \varphi(x)=C'_{1}e^{ikx}+C'_{2}e^{-ikx} 
\end{equation}
$B$H$J$k!#(B\\
$B$3$3$G(B$x=0$$B$G$N6-3&>r7o$h$j(B
\begin{equation}
 \varphi(0) = C'_{1}+C'_{2}=0 
\end{equation}
$B$7$?$,$C$F(B
\begin{equation}
\varphi(x)=C'_{1}(e^{ikx}-e^{-ikx})
=2C'_{1}i \sin{kx}
=C_{1}\sin{kx}
\end{equation}
$B$3$3$GJ?9T0\F0$7$?%0%i%U$r85$KLa$7$F!J(B$x\rightarrow x+b$$B!K(B
\begin{equation}
\varphi(x)=C_{1}\sin{k(x+b)}
\end{equation}
\item$-a<x<a$$B$N$H$-(B
$B%7%e%l%G%#%s%,!<J}Dx<0$O(B
\begin{equation}
\left\{-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}-E\right\}\varphi(x)=0
\end{equation}
$B$h$C$F(B
\begin{equation}
\left(\frac{d^{2}}{dx^{2}}-\kappa_{1}^{2}\right)\varphi(x)=0
\end{equation}
\begin{equation}
\kappa=\sqrt{\frac{2m(-E)}{\hbar^{2}}}
\end{equation}
$B$3$l$h$j(B
\begin{equation}
 \varphi(x)=C_{2}e^{\kappa x}+C_{3}e^{-\kappa x} 
\end{equation}
\item$a<x<b$$B$N$H$-(B
\end{enumerate}
$B?^#3$N%0%i%U$r(B$x$$BJ}8~$K(B-b$BJ?9T0\F0$5$;$k$H!J(B$x\rightarrow x+b$$B!K!"(B
$U(x)=-V_0$B$H$J$k(Bx$B$NHO0O$O(Ba-b<x<0$$B$H$J$k!#(B$-b<x<-a$$B$N$H$-$HF1MM$K%7%e%l%G%#%s%,!<J}Dx<0$h$j(B
\begin{equation}
\varphi(x)=C_{4}\sin{kx}
\end{equation}
$B$3$3$GJ?9T0\F0$7$?%0%i%U$r85$KLa$7$F!J(B$x\rightarrow x-b$$B!K(B
\begin{equation}
\varphi(x)=C_{4}\sin{k(x-b)}
\end{equation}\\
\\
$B0J>e$+$i(B
\[ \varphi(x) = \left\{
\begin{array}{ll}
 C_1\sin{k(x+b)}  & (-b<x<-a) \\
 C_2e^{\kappa x}+C_3e^{-\kappa x}& (-a<x<a) \\
 C_4\sin{k(x-b)}      & (a<x<b)
\end{array} \right. \]
$(-b<x<-a)$$B$K$*$$$F(B
\begin{equation}\frac{\varphi(x)}{\varphi'(x)}=\frac{\sin{k(x+b)}}{k\cos{k(x+b)}}
\end{equation}
$(-a<x<a)$$B$K$*$$$F(B
\begin{equation}\frac{\varphi(x)}{\varphi'(x)}=\frac{C_2e^{\kappa x}+C_3e^{-\kappa x}}{\kappa(C_2e^{\kappa x}-C_3e^{-\kappa x})}
\end{equation}
$(a<x<b)$$B$K$*$$$F(B
\begin{equation}\frac{\varphi(x)}{\varphi'(x)}=\frac{\sin{k(x-b)}}{k\cos{k(x-b)}}
\end{equation}
$B$3$3$G(B$x=-a,x=a$B$N$H$-(B\varphi(x)$B$*$h$S(B\varphi'(x)$$B$OO"B3$G$"$k$+$i(B

\begin{equation}
\left.\frac{\varphi(x)}{\varphi'(x)}\right|_{x=-a}=\frac{\sin{k(b-a)}}{k\cos{k(b-a)}}=\frac{C_2e^{-\kappa a}+C_3e^{\kappa a}}{\kappa(C_2e^{-\kappa a}-C_3e^{\kappa a})}\label{eq:4-6-1}
\end{equation}
\begin{equation}
\left.\frac{\varphi(x)}{\varphi'(x)}\right|_{x=a}=\frac{\sin{k(a-b)}}{k\cos{k(a-b)}}=\frac{C_2e^{\kappa a}+C_3e^{-\kappa a}}{\kappa(C_2e^{\kappa a}-C_3e^{-\kappa a})}\label{eq:4-6-2}
\end{equation}
$B$3$NFs<0$N4X78$h$j(B
\begin{equation}
\frac{C_2e^{-\kappa a}+C_3e^{\kappa a}}{\kappa(C_2e^{-\kappa a}-C_3e^{\kappa a})}=-
\frac{C_2e^{\kappa a}+C_3e^{-\kappa a}}{\kappa(C_2e^{\kappa a}-C_3e^{-\kappa a})}
\end{equation}
$B$h$C$F(B
\begin{equation}
(C_2-C_3)(C_2+C_3)=0
\end{equation}
\begin{description}
\item[A]$C_2=C_3$$B$N$H$-(B(\ref{eq:4-6-1})$B$h$j(B
\begin{equation}
\frac{\sin{k(b-a)}}{k\cos{k(b-a)}}=-\frac{\cos{\kappa a}}{\kappa \sin{\kappa a}}
\end{equation}
\begin{equation}
\frac{\kappa}{k}=-\frac{1}{\tan{k(b-a)}\tan{\kappa a}}
\end{equation}
$B$3$l$r8GM-CM$rM?$($k<0$H$9$k!#(B
\item[B]$C_2+C_3=0$$B$N$H$-(B(\ref{eq:4-6-2})$B$h$j(B
\end{description}
\begin{equation}
\frac{\sin{k(b-a)}}{k\cos{k(b-a)}}=-\frac{\sin{\kappa a}}{\kappa \cos{\kappa a}}
\end{equation}
\begin{equation}
\frac{\kappa}{k}=-\frac{\tan{\kappa a}}{\tan{k(b-a)}}
\end{equation}
$B$3$l$r8GM-CM$rM?$($k<0$H$9$k!#(B
}



