% <PLAINTEXT>
% $BNL;RJ*M}3X(B $B!'(B $B1i=,LdBj(B 4-5 $B2rEz(B
% $B:n(B  $B@.(B  $B<T(B $B!'(B f9311196 $B?e8M(B $BOB9,(B
% $B=$(B  $B@5(B  $B<T(B : g140826  $B?eLn(B $BMNJe(B
% $B:n@.G/7nF|(B $B!'(B /96/02/23
% $B=$@5G/7nF|(B : 01/12/25
% $BEE;R%a!<%k(B $B!'(B f11196@ced.cas.uec.ac.jp
% $B=$@5<TEE;R%a!<%k(B : g140826@mail.ecc.u-tokyo.ac.jp
%
\mondai
{f9311196 $B?e8M(B $BOB9,(B}
{96/02/23}
{f11196@ced.cas.uec.ac.jp}
{4-5}
{
%----$BLdBjJ8(B----%
$B<!$N%]%F%s%7%c%k%(%M%k%.!<(B$ U(x) $ ($B?^#2;2>H!K(B
\[ U(x) = \left\{
\begin{array}{ll}
 V_{1}  & (x < 0) \\
 -V_{0} & (0 < x < a) \\
 0      & (a < x)
\end{array} \right. \]
$B$NN%;6E*8GM-CM$r5a$a$k<0$N7A$r<($;!#(B\\
}
{
%----$B2rEzJ8(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-U(x))\right]
\varphi(x)=0
\end{equation}
$B$G$"$k$+$i!"GHF04X?t(B $ \varphi(x) $ $B$O<!$N$h$&$K$J$k!#(B\\

$ x<0 $  $B$N$H$-$N%7%e%l%G%#%s%,!<J}Dx<0$O!"(B
\begin{equation}
 \left(\frac{d^{2}}{dx^{2}}-\kappa_{1}^{2}\right)\varphi(x)=0
\label{eq:4-5-2}
\end{equation}
\begin{equation}
 \kappa_{1}=\sqrt{\frac{2m(V_{1}-E)}{\hbar^{2}}}
\end{equation}
$ (4-5-2) $$B<0$r2r$/$H!"(B
\begin{equation}
 \varphi(x)=B_{1}e^{\kappa_{1} x}+B_{2}e^{-\kappa_{1} x} \label{eq:4-5-4}
\end{equation}
$B$H$J$k(B( $ B_{1} $ , $ B_{2} $ $B$OG$0UDj?t(B)$B!#(B\\
$B$3$3$G6-3&>r7o(B $ \varphi(-\infty) = 0 $ $B$h$j!"(B$ B_{2}=0 $ $B!#(B $B$3$l$h$j!"(B
\begin{equation}
 \varphi(x) =  C_{1}e^{\kappa_{1} x}
\end{equation}
$B$H$J$k(B $ (C_{1}=B_{1}) $$B!#(B\\

$ 0 < x < a $ $B$N$H$-$N%7%e%l%G%#%s%,!<J}Dx<0$O!"(B
\begin{equation}
 \left( \frac{d^{2}}{dx^{2}} + k^{2} \right) \varphi(x) = 0 \label{eq:4-5-6}
\end{equation}
\begin{equation}
 k=\sqrt{\frac{2m(V_{0}+E)}{\hbar^{2}}}
\end{equation}
$ (4-5-6) $ $B<0$r2r$/$H!"(B
\begin{equation}
 \varphi(x) = C_{2}e^{ikx} + C_{3}e^{-ikx} \label{eq:4-5-8}
\end{equation}
$B$H$J$k(B ($ C_{2} $ , $ C_{3} $ $B$OG$0UDj?t(B)$B!#(B\\

$B:G8e$K(B $ a<x $  $B$N$H$-$N%7%e%l%G%#%s%,!<J}Dx<0$O!"(B
\begin{equation}
 \left(\frac{d^{2}}{dx^{2}}-\kappa^{2}\right)\varphi(x)=0 \label{eq:4-5-9}
\end{equation}
\begin{equation}
 \kappa=\sqrt{\frac{2m(-E)}{\hbar^{2}}}
\end{equation}
$ (4-5-9) $$B<0$r2r$/$H!"(B
\begin{equation}
 \varphi(x)=B_{3}e^{\kappa x}+B_{4}e^{-\kappa x} \label{eq:4-5-11}
\end{equation}
$B$H$J$k(B( $ C_{3} $ , $ C_{4} $ $B$OG$0UDj?t(B)$B!#(B\\
$B$3$3$G6-3&>r7o(B $ \varphi(\infty) = 0 $ $B$h$j!"(B$ B_{3}=0 $ $B!#(B $B$3$l$h$j!"(B
\begin{equation}
 \varphi(x) =  C_{4}e^{-\kappa x}
\end{equation}
$B$H$J$k(B $ (B_{4}=C_{4}) $$B!#(B\\

$B$^$?!"(B$ x=0 $ $B$N$H$-(B $ \varphi(0) $ $B$*$h$S(B $ \varphi '(0) $ $B$OO"B3$G$"$k$+(B
$B$i!"(B
\begin{equation}
 \displaystyle \left. \frac{\varphi(x)}{\varphi'(x)} \right|_{x=0} =
 \kappa_{1}^{-1}\hspace{5mm}(x<0)
\end{equation}
\begin{equation}
 \displaystyle \left. \frac{\varphi(x)}{\varphi'(x)} \right|_{x=0} =
 \frac{1 + A}{ik(1 - A)}\hspace{5mm}(0<x<a)
\end{equation}
$B$H$J$j(B( $ C_{3}/C_{2} = A $ )$B!"(B
\begin{equation}
 \displaystyle \kappa_{1} = i k \frac{1-A}{1+A}
\end{equation}
$B$,F@$i$l$k!#(B \\
$BF1$8$/!"(B$ x=a $ $B$N$H$-(B $ \varphi(a) $ $B$*$h$S(B $ \varphi '(a) $
$B$OO"B3$G$"$k$+$i!"(B
\begin{equation}
 \displaystyle \left.\frac{\varphi(x)}{\varphi'(x)} \right|_{x=a} =
\frac{e^{ika} + Ae^{-ika}}{ik(e^{ika} - Ae^{-ika})}\hspace{5mm}(0<x<a)
\end{equation}
\begin{equation}
 \displaystyle \left. \frac{\varphi(x)}{\varphi'(x)} \right|_{x=a} =
 -\kappa^{-1}\hspace{5mm}(x>a)
\end{equation}
$B$H$J$j(B( $ C_{3}/C_{2} = A $ )$B!"(B
\begin{equation}
 \displaystyle -\kappa = i k \frac{e^{ika} - Ae^{-ika}}{e^{ika} +Ae^{-ika}}
\end{equation}
$B$,F@$i$l$k!#$3$l$h$j!"(B\\
\begin{equation}
 \displaystyle \frac{\kappa}{k} =
  - \frac{\kappa_{1} - k \tan (ka)}
         {k + \kappa_{1} \tan(ka)}
\end{equation}
$B$,F@$i$l$k!#(B
}
