% <PLAINTEXT>
% $BNL;RJ*M}3X(B $B!'(B $B1i=,LdBj(B 4-1 $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 : /96/02/19
% $B=$@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 $BOB9,(B}
{96/02/19}
{f11196@ced.cas.uec.ac.jp}
{4-1}
{
%----$BLdBjJ8(B----%
$B<!$N%]%F%s%7%c%k%(%M%k%.!<(B$ U(x) $ ($B?^#1;2>H!K(B
\[ U(x) = \left\{
\begin{array}{ll}
 \infty & (x < 0) \\
 -V_{0} & (0 < x < a) \\
 0      & (a < x)
\end{array} \right. \]
\\
$B$NB+G{$5$l$?%(%M%k%.!<(B$ (E<0) $$B$r5a$a$h!#(B
}
{
%----$B2rEzJ8(B---%
$B%7%e%l%G%#%s%,!<J}Dx<0$O!"(B
\begin{equation}
 H\varphi(x)=E\varphi(x) \label{eq:4-1-2}
\end{equation}
$B$G$"$k!#$3$3$G!"(B
\begin{equation}
 H=\frac{p^{2}_{x}}{2m}+U(x)
\end{equation}
\begin{equation}
 p_{x} = -i\hbar\frac{d}{dx}
\end{equation}
$B$G$"$k$+$i!"(B$ (4-1-1) $$B<0$O!"(B
\begin{equation}
 \left[-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}-(E-U(x))\right]
\varphi(x)=0
\end{equation}
$B$H$J$k!#$3$l$h$j0f8M$KB+G{$5$l$?%(%M%k%.!<(B$ (E<0) $$B$r5a$a$k!#(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:4-1-5}
\end{equation}
\begin{equation}
 k=\sqrt{\frac{2m(V_{0}+E)}{\hbar^{2}}}
\end{equation}
$ (4-1-5) $ $B<0$r2r$/$H!"(B
\begin{equation}
 \varphi(x) = C_{1}e^{ikx} + C_{2}e^{-ikx} \label{eq:4-1-7}
\end{equation}
$B$H$J$k(B ($ C_{1} $ , $ C_{2} $ $B$OG$0UDj?t(B)$B!#(B\\
$B$3$3$G6-3&>r7o(B $ \varphi(0)=0 $ $B$h$j!"(B$ C_{1}=-C_{2} $ $B!#(B $B$3$l$h$j!"(B
\begin{equation}
\begin{array}{lcl}
 \varphi(x) & = & C_{1}e^{ikx}-C_{1}e^{-ikx} \\
            & = & 2iC_{1}\sin(kx) \\
            & = & A\sin(kx)
\end{array}
\end{equation}
$B$H$J$k(B $ (A=2iC_{1}) $ $B!#(B

$B<!$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:4-1-9}
\end{equation}
\begin{equation}
 \kappa=\sqrt{\frac{2m(-E)}{\hbar^{2}}}
\end{equation}
$ (4-1-9) $$B<0$r2r$/$H!"(B
\begin{equation}
 \varphi(x)=C_{3}e^{\kappa x}+C_{4}e^{-\kappa x} \label{eq:4-1-11}
\end{equation}
$B$H$J$k(B( $ C_{3} $ , $ C_{4} $ $B$OG$0UDj?t(B)$B!#(B\\
$B$3$3$G6-3&>r7o(B $ \varphi(\infty) = 0 $ $B$h$j!"(B$ C_{3}=0 $ $B!#(B $B$3$l$h$j!"(B
\begin{equation}
 \varphi(x) =  Be^{-\kappa x}
\end{equation}
$B$H$J$k(B $ (B=C_{4}) $$B!#(B\\
$B$^$?!"(B$ x=a $ $B$N$H$-(B $ \varphi(a) $ $B$*$h$S(B $ \varphi '(a) $ $B$OO"B3$G$"$k$+(B
$B$i!"(B
\begin{equation}
 \frac{\varphi(x)}{\varphi'(x)}=k^{-1}\tan(ka)\hspace{5mm}(0<x<a)
\end{equation}
\begin{equation}
 \frac{\varphi(x)}{\varphi'(x)}=-\kappa^{-1}\hspace{5mm}(x>a)
\end{equation}
$B$3$l$h$j!"(B$ \displaystyle \tan(ka) = -\frac{k}{\kappa} $ $B$,F@$i$l$k!#(B
}







