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% $B:n@.G/7nF|(B : /02/12/10
% $BEE;R%a!<%k(B : micchan@bf7.so-net.ne.jp
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%  \mondai{$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
{g240947 $B0f>eF;M:(B}
{02/12/10}
{micchan@bf7.so.net.ne.jp}
{7-6}
{$B?^$N%]%F%s%7%c%k$N>l9g$K$D$$$F!"(B(7.26)$B$GDj5A$7$?E>Aw9TNs(B$M$$B$NI=<0$r5a$a$h!#(B
($BI,MW$K1~$8$F!"(B(7.24)$B$GDj5A$7$?9TNs(B$Q\left(x\right)$,$K\left(x\right)$$B$rDj5A(B
$B$;$h!#9TNs$rE83+$9$kI,MW$O$J$$!K!#(B}
{
$B$^$:!"0J2<$N$=$l$>$l$N>u67$G$NGHF04X?t$r5a$a$F$*$/!#(B(A$B!A(BE$B$ODj?t(B)$B!!(B\\
(1)$B:8C<$H>cJI$K64$^$l$?HO0O!!(B\\
$B1&$K?J9T$9$kGH$H%]%F%s%7%c%k>cJI$G$NH?<MGH$,B8:_$9$k$N$G!"(B
\begin{eqnarray}
\varphi_{1}=Ae^{ikx}+Be^{-ikx} \label{eq:Q7-6-1}
\end{eqnarray}
(2)$B>cJICf(B
\begin{eqnarray}
\varphi_{2}=Ce^{\kappa x}+De^{-\kappa x} \label{eq:Q7-6-2}
\end{eqnarray}
(3)$B1&C<(B \\
$BH?<MGH$,B8:_$7$J$$$N$G!"(B
\begin{eqnarray}
\varphi_{3}=Ee^{ikx} \label{eq:Q7-6-3}
\end{eqnarray}$B!!(B
$B$3$3$G!"(B
\begin{eqnarray}
Q\left(x\right) &=&
\left(
\begin{array}{cc}
e^{ikx} & e^{-ikx} \\
ye^{ikx} & -ye^{-ikx} \\
\end{array}
\right) \label{eq:Q7-6-4}
\end{eqnarray}
\begin{eqnarray}
K\left(x\right) &=&
\left(
\begin{array}{cc}
e^{\kappa x} & e^{-\kappa x} \\
e^{\kappa x} & -e^{-\kappa x} \\
\end{array}
\right)\label{eq:Q7-6-5}
\end{eqnarray}
\begin{eqnarray}
K^{'}\left(x\right) &=&
\left(
\begin{array}{cc}
e^{\kappa^{'}x} & e^{-\kappa^{'}x} \\
e^{\kappa^{'}x} & -e^{-\kappa^{'}x} \\
\end{array}
\right)\\$B!!(B\label{eq:Q7-6-6}
J\left(x\right) &=& Q\left(x\right)^{-1}K\left(x\right) \\ \label{eq:Q7-6-7}
J^{'}\left(x\right) &=& Q\left(x\right)^{-1}K^{'}\left(x\right)
\label{eq:Q7-6-8}
\end{eqnarray}
$B$HDj5A$7!"$=$l$>$l!"(B
\[
k = \sqrt{\frac{2mE}{\hbar}} ,
\kappa = \frac{\sqrt{2m\left(V_{0}-E\right)}}{\hbar} ,
\kappa^{'} = \frac{\sqrt{2m\left(V_{0}/3-E\right)}}{\hbar} ,
y = \frac{ik}{\kappa}
\]
$B$G$"$j!"(B$E$$B$OEE;R$N;}$D%(%M%k%.!<(B , $m$$B$OEE;R$N<ANL(B $B$r$"$i$o$9!#(B

$B$^$?!"$=$l$>$l$NHO0O$N6-3&$r4^$`Fs$D$NGHF04X?t$NCM$H!"HyJ,$OEy$7$$$O$:$G$"$k(B
$B$N$G!"(B
$B<!$NFs$D$N>l9g$K$*$$$F!"$=$l$>$lJ}Dx<0$,@.$jN)$D!#(B\\
(1)$B>cJI$KF~$k>l9g!J$3$N$H$-$N(B$x$$B:BI8$r(B$x=s$$B$H$9$k!K(B
\begin{eqnarray}
Ae^{iks} + Be^{-iks} &=& C e^{\kappa s}+ D e^{-\kappa s} \\
\label{eq:Q7-6-9}
ik\left(Ae^{iks} - Be^{-iks}\right) &=& \kappa\left(C e^{\kappa s}+ D
e^{-\kappa s}\right) \label{eq:Q7-6-10}
\end{eqnarray}

$B$3$l$r9TNs$GI=$9$H!"(B
\begin{eqnarray}
\left(
\begin{array}{cc}
e^{iks} & e^{-iks} \\
ye^{iks} & -ye^{-iks} \\
\end{array}
\right)
\left(
\begin{array}{c}
A \\
B \\
\end{array}
\right)
&=&
\left(
\begin{array}{cc}
e^{\kappa s} & e^{-\kappa s} \\
e^{\kappa s} & -e^{-\kappa s} \\
\end{array}
\right)
\left(
\begin{array}{c}
C \\
D \\
\end{array}
\right) \label{eq:Q7-6-11}
\end{eqnarray}
$B$3$l$rJQ7A$9$k$H!"<0(B(\ref{eq:Q7-6-4})$B!"<0(B(\ref{eq:Q7-6-5})$B$h$j(B
\begin{eqnarray}
\left(
\begin{array}{c}
A \\
B \\
\end{array}
\right)
&=&
Q\left(s\right)^{-1}K\left(s\right)
\left(
\begin{array}{c}
C \\
D \\
\end{array}
\right)\\
&=&
J\left(s\right)
\left(
\begin{array}{c}
C \\
D \\
\end{array}
\right) \label{eq:Q7-6-12}
\end{eqnarray}
$B$H$J$j!">cJI$KF~$k>l9g$O!"(B$J\left(x\right)$$B$r:nMQ$5$;$l$P$$$$$3$H$,$o$+$k!#(B
\\
(2)$B>cJI$+$i=P$k>l9g!J$3$N$H$-$N(B$x$$B:BI8$r(B$x=t$$B$H$9$k(B)
\begin{eqnarray}
 C e^{\kappa t}+ D e^{-\kappa t} &=& A^{'}e^{ikt} + B^{'}e^{-ikt} \\$B!!(B
\label{eq:Q7-6-13}
\kappa\left(C e^{\kappa t}+ D e^{-\kappa t}\right) &=&
ik\left(A^{'}e^{ikt} - B^{'}e^{-ikt}\right) \label{eq:Q7-6-14}
\end{eqnarray}
$BF1MM$K$7$F9TNs$GI=$9$H!"(B
\begin{eqnarray}
\left(
\begin{array}{cc}
e^{\kappa t} & e^{-\kappa t} \\
e^{\kappa t} & -e^{-\kappa t} \\
\end{array}
\right)
\left(
\begin{array}{c}
C \\
D \\
\end{array}
\right)
&=&
\left(
\begin{array}{cc}
e^{ikt} & e^{-ikt} \\
ye^{ikt} & -ye^{-ikt} \\
\end{array}
\right)
\left(
\begin{array}{c}
A^{'} \\
B^{'} \\
\end{array}
\right) \label{eq:Q7-6-15}
\end{eqnarray}
$B$3$l$rJQ7A$9$k$H!"<0(B(\ref{eq:Q7-6-4})$B!"(B($B<0(B\ref{eq:Q7-6-5})$B$h$j(B
\begin{eqnarray}
\left(
\begin{array}{c}
C \\
D \\
\end{array}
\right)
&=&
K\left(t\right)^{-1}Q\left(t\right)
\left(
\begin{array}{c}
A^{'} \\
B^{'} \\
\end{array}
\right)\\
&=&
J\left(t\right)^{-1}
\left(
\begin{array}{c}
A^{'} \\
B^{'} \\
\end{array}
\right) \label{eq:Q7-6-16}
\end{eqnarray}
$B$H$J$j!">cJI$+$i=P$k>l9g$O!"(B$J\left(x\right)^{-1}$$B$r:nMQ$5$;$l$P$h$/!"(B
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$B9g$K$bF1MM$K$9$l$P$$$$$H$o$+$k!#(B
$B0J>e$O%]%F%s%7%c%k$,(B$V_{0}$$B$N>l9g$G$"$j!"%]%F%s%7%c%k$,(B$V_{0}/3$$B$N>l9g$bF1MM(B
$B$K9M$($k$H!"(B
$B>cJI$KF~$k$H$-$O(B$J^{'}\left(x\right)$$B$r:nMQ$5$;!"(B
$B>cJI$r=P$k$H$-$O(B$J^{'-1}\left(x\right)$$B$r:nMQ$5$;$l$P$h$$$3$H$,$o$+$k!#$3$3(B
$B$G:G8e$K$G$-$?(B
$B9TNs$N@Q$r(B$M$$B$HDj5A$7$?!#$7$?$,$C$FEz$($O!"(B\\
(a)
\begin{eqnarray}
M &=&
J\left(0\right)J^{-1}\left(a\right)J\left(3a\right)J^{-1}\left(4a\right)
\label{eq:Q7-6-17}
\end{eqnarray}
(b)
\begin{eqnarray}
M &=&
J\left(0\right)J^{-1}\left(a\right)J\left(2a\right)J^{-1}\left(4a\right)
\label{eq:Q7-6-18}
\end{eqnarray}
(c)
\begin{eqnarray}
M &=&
J\left(0\right)J^{-1}\left(a\right)J\left(2a\right)J^{-1}\left(3a\right)
J\left(4a\right)J^{-1}\left(5a\right)$B!!!!!!(B//$B$3$3$G4V0c$C$?2~9T$,5/$3$C$F$$(B
$B$^$7$?!#(B
\label{eq:Q76-19}
\end{eqnarray}
(d)
\begin{eqnarray}
M &=&
J\left(0\right)J^{'-1}\left(a\right)J^{'}\left(3a\right)J^{-1}
\left(4a\right) \label{eq:Q7-3-18}
\end{eqnarray}
$B$H$J$k!#(B}

