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% $BNL;RJ*M}3X(B $B!'(B $B1i=,LdBj(B 7-5 $B2rEz(B
% $B:n(B  $B@.(B  $B<T(B $B!'(B g040496 $B>BED(B $BOB9,(B
% $B:n@.G/7nF|(B $B!'(B 01/02/06
% $BEE;R%a!<%k(B $B!'(B g040496@mail.ecc.u-tokyo.ac.jp
%
\mondai
{g040496 $B>BED(B $BOB9,(B}
{01/01/30}
{g040496@mail.ecc.u-tokyo.ac.jp}
{7-5}
{ 
%----$BLdBjJ8(B----%
(7.24)$B<0$GDj5A$5$l$?!"#29T#2Ns$N9TNs(B$\displaystyle Q(x)$$B$rMQ$$$F!"#1<!85>cJI$,#18D$NLdBj(B(7.1)$B$r2r$1!#(B(7.27)$B$rMQ$$$?7k2L$O!"(B(7.13)$B$KEy$7$/$J$k$3$H$r3NG'$;$h!#(B
} 
{ 
%----$B2rEz(B----%
(7.24)$B<0$h$j(B$\displaystyle Q(x)$B!"(BK(x)$$B$O0J2<$GDj5A$5$l$k!#(B
\begin{equation} \displaystyle
	Q(x)= \left(  \begin{array}{cc}
		e^{ikx} & e^{-ikx} \\
		ye^{ikx} & -ye^{-ikx}
	\end{array} \right) ,\ \ 
	K(x)= \left(  \begin{array}{cc}
		e^{\kappa x} & e^{-\kappa x} \\
		e^{\kappa x} & -e^{-\kappa x}
	\end{array} \right) 
\end{equation}
$BC"$7(B \ $ y= ik/\kappa \ , \ k=\sqrt[]{2mE}/ \hbar \ , \ \kappa=\sqrt[]{2m(V_0-E)}/ \hbar$ $B$H$*$$$?!#(B\\
$B%]%F%s%7%c%k$GJ,3d$5$l$?3FNN0h$NGHF04X?t$r$=$l$>$l(B $\varphi_1\ , \ \varphi_2\ , \ \varphi_3$$B$H$9$l$P!"(B
\begin{eqnarray} \displaystyle
	\varphi_1 &=&  \ \  e^{ikx}+Ae^{-ikx} \ \ ( x \leq -a/2 ) \ , \ \nonumber \\
	\varphi_2 &=& Be^{\kappa x}+Ce^{-\kappa x} \ \ ( -a/2 \leq x \leq a/2 ) \ , \ \nonumber \\
	\varphi_3 &=& De^{ikx} \ \ ( a/2 \leq x ) \nonumber
\end{eqnarray}
$B$3$l$h$j6-3&>r7o$O(B
\begin{eqnarray} \displaystyle
	e^{-ika/2}+Ae^{ika/2} &=& Be^{-\kappa a/2}+Ce^{\kappa a/2} \nonumber\\
	ik\left( e^{-ika/2}-Ae^{ika/2} \right) &=& \kappa \left( Be^{-\kappa a/2}-$B#C(Be^{\kappa a/2} \right) \nonumber \\
	Be^{\kappa a/2}+Ce^{-\kappa a/2} &=& De^{ika/2} \nonumber\\
	\kappa \left( Be^{\kappa a/2}-Ce^{-\kappa a/2} \right) &=& ikDe^{ika/2} \nonumber
\end{eqnarray}
$B$3$l$r9TNs$N7A$K=q$-D>$;$P!"(B
\begin{eqnarray} \displaystyle
	\left( \begin{array}{cc}
		e^{-ika/2} & e^{ika/2} \\
		(ik/\kappa)e^{-ika/2} & -(ik/\kappa)e^{ika/2}
	\end{array} \right)
	\left( \begin{array}{c} 1 \\ A \end{array} \right)
	&=&
	\left( \begin{array}{cc}
		e^{-\kappa a/2} & e^{\kappa a/2} \\
		e^{-\kappa a/2} & -e^{\kappa a/2} 
	\end{array} \right)
	\left( \begin{array}{c} B \\ C \end{array} \right) \nonumber \\
	\left( \begin{array}{cc}
		e^{\kappa a/2} & e^{-\kappa a/2} \\
		e^{\kappa a/2} & -e^{-\kappa a/2}
	\end{array} \right)
	\left( \begin{array}{c} B \\ C \end{array} \right)
	&=&
	\left( \begin{array}{cc}
		e^{ika/2} & e^{-ika/2} \\
		(ik/\kappa) e^{ika/2} & -(ik/\kappa) e^{-ika/2} 
	\end{array} \right)
	\left( \begin{array}{c} D \\ 0 \end{array} \right) \nonumber
\end{eqnarray}
	$B$3$3$G(B$\displaystyle Q(x)$B!"(BK(x)$$B5Z$S(B\ $ y= ik/\kappa $ \  $B$rF3F~$9$k$H(B
\begin{eqnarray}
	Q(-a/2)
	\left( \begin{array}{c} 1 \\ A \end{array} \right)
	&=&
	K(-a/2)
	\left( \begin{array}{c} B \\ C \end{array} \right) \nonumber \\
	K(a/2)
	\left( \begin{array}{c} B \\ C \end{array} \right)
	&=&
	Q(a/2)
	\left( \begin{array}{c} D \\ 0 \end{array} \right) \nonumber 
\end{eqnarray}
$B99$K(B\ $\displaystyle J(x)=Q(x)^{-1}K(x)$\ $B$H$*$/$H(B
\begin{eqnarray} \displaystyle
	\left( \begin{array}{c} 1 \\ A \end{array} \right) &=&
	Q(-a/2)^{-1}K(-a/2) \left( \begin{array}{c} B \\ C \end{array} \right) \nonumber \\
	&=&Q(-a/2)^{-1}K(-a/2)Q(a/2)^{-1}K(a/2) \left( \begin{array}{c} D \\ 0 \end{array} \right) \nonumber \\
	&=&J(-a/2)J(a/2)^{-1} \left( \begin{array}{c} D \\ 0 \end{array} \right) \nonumber
\end{eqnarray}
$B=>$C$FE>Aw9TNs$r(B$\displaystyle M=(M_{ij})=J(-a/2)J(a/2)^{-1}$$B$H$9$l$P(B
\begin{equation} \displaystyle
	\left( \begin{array}{c} 1 \\ A \end{array} \right) =
	\left( \begin{array}{cc} M_{11} & M_{12} \\ M_{21} & M_{22}
	\end{array} \right) \left( \begin{array}{c} D \\ 0 \end{array} \right) 
	= \left( \begin{array}{c} M_{11}D \\ M_{21}D \end{array} \right) 
\end{equation}
$B$G$"$k$+$i(B \ \ $\displaystyle T=\left|D\right|^2=\left|M_{11}^{-1}\right|^2$ \ \ $B$r5a$a$k$3$H$O(B\ \ $\displaystyle J(-a/2)J(a/2)^{-1}$$B$N(B(1,1)$B@.J,$r5a$a$k;v$K5"Ce$5$l$k!#$^$:(B$\displaystyle J(x)$$B$r5a$a$k!#(B
\begin{eqnarray}
	\left|Q(x)\right| &=& 
	\left| \begin{array}{cc}
		e^{ikx} & e^{-ikx} \\
		ye^{ikx} & -ye^{-ikx}
	\end{array} \right| \nonumber\\
	&=& e^{ikx}(-ye^{-ikx})-e^{-ikx}ye^{ikx} \nonumber\\
	&=& -2y \nonumber \\ \nonumber\\
	$B!h(B \ \  Q(x)^{-1} &=& \frac{1}{2y}
	\left( \begin{array}{cc}
		ye^{-ikx} & e^{-ikx} \\
		ye^{ikx} & -e^{ikx}
	\end{array} \right) \nonumber \\ \nonumber \\
	$B!h(B \ \ J(x)=Q(x)^{-1}K(x) &=& \frac{1}{2y}
	\left( \begin{array}{cc}
		ye^{-ikx} & e^{-ikx} \\
		ye^{ikx} & -e^{ikx}
	\end{array} \right)
	\left(  \begin{array}{cc}
		e^{\kappa x} & e^{-\kappa x} \\
		e^{\kappa x} & -e^{-\kappa x}
	\end{array} \right) \nonumber\\
	&=& \frac{1}{2y}
	\left( \begin{array}{cc}
		(y+1)e^{(-ik+\kappa)x}&
		(y-1)e^{(-ik-\kappa)x} \\
		(y-1)e^{(ik+\kappa)x}&
		(y+1)e^{(ik-\kappa)x} 
	\end{array} \right) \nonumber 
\end{eqnarray}	
$BB3$$$F(B$\displaystyle J(x)^{-1}$$B$r5a$a!":G8e$K(B$\displaystyle J(-a/2)J(a/2)^{-1}$$B$r7W;;$9$k!#(B
\begin{eqnarray}
	\left|J(x)\right| &=& \frac{1}{(2y)^2} \left\{
	(y+1)e^{(-ik+\kappa)x}(y+1)e^{(ik-\kappa)x} - (y-1)e^{(-ik-\kappa)x}(y-1)e^{(ik+\kappa)x} \right\} \nonumber \\
	&=& \frac{1}{4y^2} \left\{ (y+1)^2 - (y-1)^2 \right\} \nonumber \\
	&=& \frac{1}{y} \nonumber\\
	$B!h(B \ \ J(x)^{-1} &=& -\frac{1}{2}
	\left( \begin{array}{cc}
		(-y-1)e^{(ik-\kappa)x} &
		(y-1)e^{(-ik-\kappa)x} \\
		(y-1)e^{(ik+\kappa)x} &
		(-y-1)e^{(-ik+\kappa)x}
	\end{array} \right) \nonumber 
\end{eqnarray}
$B0J>e$h$j(B$\displaystyle M_{11}=J(-a/2)J(a/2)^{-1}$$B$O(B
\begin{eqnarray}
	& & -\frac{1}{4y} \left\{ 
		(y+1)e^{(-ik+\kappa)(-a/2)}(-y-1)e^{(ik-\kappa)(a/2)}
		- (y-1)e^{(-ik-\kappa)(-a/2)}(y-1)e^{(ik+\kappa)(a/2)} \right\} \nonumber \\
	&=& \frac{e^{ika}}{4y} \left\{
		(y+1)^2e^{-\kappa a} - (y-1)^2e^{\kappa a} \right\} \nonumber \\
	&=& \frac{e^{ika}}{2y} \left\{
		-(y^2+1) \frac{ \left( e^{\kappa a} - e^{-\kappa a} \right) }{2}
		+2y \frac{ \left( e^{\kappa a} + e^{-\kappa a} \right) }{2} 
		\right\} \nonumber \\
	&=& \frac{e^{ika}}{2y} \left\{
		-(y^2+1) \sinh(\kappa a) + 2y \cosh(\kappa a) \right\} 
\end{eqnarray}
$B$H$J$j!"$3$N@dBPCM$N5U?t$N#2>h$,(B$T$$B$H$J$k!#(B
\begin{eqnarray}
	y^*=-y \ \ \left( \ $B"h(B\ y=i\frac{k}{\kappa} \ \right)& & \nonumber \\
	\cosh^2(x)-\sinh^2(x)=1 \ & &\nonumber
\end{eqnarray}
$B$rMQ$$$F(B$\displaystyle \left| -(y^2+1) \sinh(\kappa a) + 2y \cosh(\kappa a) \right|^2 = \left| Q \right| ^2 /4$$B$r3N$+$a$k!#(B
\begin{eqnarray}
	& & \left| -(y^2+1) \sinh(\kappa a) + 2y \cosh(\kappa a) \right|^2 \nonumber \\
	&=& \left\{ -(y^2+1) \sinh(\kappa a) + 2y \cosh(\kappa a) \right\} \left\{ -(y^2+1) \sinh(\kappa a) + 2y \cosh(\kappa a) \right\} \nonumber \\
	&=& (y^2+1)^2\sinh^2(\kappa a)-4y^2\cosh^2(\kappa a) \nonumber \\
	&=& \left| Q \right| ^2 /4 \nonumber
\end{eqnarray}
$B$h$C$F(B
\begin{equation}
	T=\frac{-4y^2}{\left|Q\right|^2/4}=\frac{1}{\left|Q\right|^2}\frac{16k^2}{\kappa^2} 
\end{equation}
$B$G$"$j!"7k2L$O(B(7.13)$B$HEy$7$$!#(B
}



