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% $B:n(B  $B@.(B  $B<T(B : g140828 $BIp@n!!=c(B
% $B:n@.G/7nF|(B : /01/12/25
% $BEE;R%a!<%k(B : g140828@mail.ecc.u-tokyo.ac.jp
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% \mondai{$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
{g140828 $BIp@n!!=c(B}
{01/12/25}
{g140828@mail.ecc.u-tokyo.ac.jp}
{4-2}
{$B?^#1$N%]%F%s%7%c%k$G(B$E > 0$$B$N8GM-4X?t$N7A$r5a$a$h!#$?$@$7!"5,3J2=$O$7$J$/$F(B
$B$h$$!#(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 - U(x))} \right]\varphi
(x) = 0
\end{equation}
$B$G$"$k$+$i!"8GM-4X?t$O<!$N$h$&$K$J$k!#(B\\
$x < 0$$B$N$H$-$OEE;R$OB8:_$7$J$$$H9M$($k!#(B
$0 < x < a$$B$N$H$-$N%7%e%l%G%#%s%,!<J}Dx<0$O!"(B
\begin{eqnarray}
 \left( {\frac{d^2}{dx^2} + k^2} \right)\varphi (x) = 0 \\
 k = \sqrt {\frac{2m(V_0 + E)}{\hbar ^2}}
\end{eqnarray}
$B$3$l$r2r$-!"(B
\begin{equation}
\varphi (x) = C_1 e^{ikx} + C_2 e^{ - ikx}
\end{equation}
$B$H$J$k!J(B$C_1 ,C_2 $$B$OG$0UDj?t!K!#(B
$B$3$3$G6-3&>r7o(B$\varphi (0) = 0$$B$h$j!"(B$C_1 = - C_2 $$B!#$3$l$h$j!"(B
\begin{eqnarray}
 \varphi (x) &=& C_1 e^{ikx} - C_1 e^{ - ikx} \nonumber \\
 &=& 2iC_1 \sin (kx) \nonumber \\
 &=& A\sin (kx)
\end{eqnarray}
$B$H$J$k!J(B$A = 2iC_1 $$B!K!#(B\\
$a < x$$B$N$H$-$N%7%e%l%G%#%s%,!<J}Dx<0$O!"(B
\begin{eqnarray}
 \left( {\frac{d^2}{dx^2} + q^2} \right)\varphi (x) = 0 \\
 q = \sqrt {\frac{2mE}{\hbar ^2}}
 \end{eqnarray}
$B$3$l$r2r$-!"(B
\begin{equation}
\varphi (x) = C_3 e^{iqx} + C_4 e^{ - iqx}
\end{equation}
$B$H$J$k!J(B$C_3 ,C_4 $$B$OG$0UDj?t!K!#(B
$B$^$?!"(B$x = a$$B$N$H$-(B$\varphi (x)$$B$*$h$S(B$\varphi '(x)$$B$OO"B3$G$"$k$+$i!"(B
\begin{eqnarray}
A\sin (ka) &=& C_3 e^{iqa} + C_4 e^{ - iqa}\\
kA\cos (ka) &=& iqC_3 e^{iqa} - iqC_4 e^{ - iqa}
\end{eqnarray}
$B$3$N#2<0$h$j!"(B
\begin{eqnarray}
 C_3 &=& \frac{\left\{iq\sin (ka) + k\cos (ka)\right\}e^{ - iqa}}{2iq}A \\
 C_4 &=& \frac{\left\{iq\sin (ka) - k\cos (ka)\right\}e^{iqa}}{2iq}A
\end{eqnarray}
$B$,F@$i$l$k!#$3$l$h$j8GM-4X?t$O(B$0 < x < a$$B$N$H$-!"(B
\begin{equation}
\varphi (x) = A\sin (kx)
\end{equation}
$a < x$$B$N$H$-!"(B
\begin{eqnarray}
\varphi (x) = \frac{\left\{iq\sin (ka) + k\cos (ka)\right\}e^{ -
iqa}}{2iq}Ae^{iqx} \nonumber \\
+ \frac{\left\{iq\sin (ka) - k\cos (ka)\right\}e^{iqa}}{2iq}Ae^{ - iqx}
\end{eqnarray}
$B$H$J$k!#(B
}


