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% $B=$!!@5!!<T(B $B!'(B 241025B $B;32<!!??(B
% $B:n@.G/7nF|(B $B!'(B /02/10/14
% $B=$@5G/7nF|(B $B!'(B /02/12/5
% $BEE;R%a!<%k(B $B!'(B g240916@mail.ecc.u-tokyo.ac.jp
% $B=$@5<TEE;R%a!<%k(B $B!'(B g241025@mail.ecc.u-tokyo.ac.jp
%
% \mondai{$B:n@.<T(B}{$B:n@.G/7nF|(B}{$BEE;R%a!<%k(B}{$BLdBjHV9f(B}{$BLdBjJ8(B}{$B2rEz(B}
%
\mondai
{g241025 $B;32<(B $B??(B}
{02/12/4}
{g241025@mail.ecc.u-tokyo.ac.jp}
{5-5}
{
%----$BLdBjJ8(B----%

$BIT3NDj@-86M}$rMQ$$$F!"<!$N%]%F%s%7%c%k(B$U(x)$$B$N>l9g$NEE;R$N4pDl>uBV$N%(%M(B
$B%k%.!<$rI>2A$;$h!#$^$?(B$(a)-(c)$$B$K$*$$$F%S%j%"%kDjM}$O@.N)$9$k$+!#(B
\begin{enumerate}
\item[(a)]$U(x)=\displaystyle{-\frac{1}{|x|}C}$
\item[(b)]$U(x)=|x|C$
\item[(c)]$U(x)=x^4C$
\item[(b)]$U(x)=-e^{-ax^2}C$B!JCm!'(B\displaystyle{\frac{a\hbar^2}{8mC}}\ll1
$B$H$;$h!#!K(B$
\end{enumerate}
}
{
%----$B2rEzJ8(B---%

$BIT3NDj@-86M}$h$j!"4pDl>uBV$N%(%M%k%.!<(B$E$$B$O(B
\begin{equation}
E=\frac{p^2}{2m}+U(x)\sim\frac{\hbar^2}{8mx^2}+U(x)
\end{equation}
$B$H=q$1$k!#(B\\
\begin{description}
\item[(a)]
\begin{equation}
E=\frac{\hbar^2}{8mx^2}-\frac{1}{|x|}C
\end{equation}
$B$H$J$k!#(B\\
$B$3$l$O(B$x$$B<4BP>N$@$+$i(B$0<x$$B$N>l9g$G9M$($l$PNI$$!#(B
\begin{equation}
\frac{dE}{dx}=-\frac{1}{4mx^3}(\hbar^2-4mCx)
\end{equation}
$B$H$J$k!#(B\\
$B=>$C$F0J2<$NA}8:I=$rF@$k!#(B
\\
\\
\begin{tabular}{c|rrrrr}
$x$&0&$\qquad$&$\frac{\hbar^2}{4mC}$&$\qquad$&$\infty$\\ \hline
$\frac{dE}{dx}$&$\qquad$&$-$&$0$&+&0\\  \hline
$E$&$\qquad$&$\searrow$&$\qquad$&$\nearrow$&0\\
\end{tabular}\\
\\
\\
$E$$B$,:G>.$H$J$k$N$O!"A}8:I=$h$j(B\\
$x=\displaystyle{\frac{\hbar^2}{4mC}}$$B$N$H$-$G!"$3$N$H$-(B
\begin{equation}
E=T+U=\frac{2mC^2}{\hbar^2}-\frac{4mC^2}{\hbar^2}=-\frac{2mC^2}{\hbar^2}
\end{equation}
$B$^$?!"(B$T:U=-1:2$$B$G$"$j!"$3$l$O%S%j%"%kDjM}$rK~$?$9!#(B
\item[(b)]
\begin{equation}
E=\frac{\hbar^2}{8mx^2}+|x|C
\end{equation}
$B$H$J$k!#(B\\$\qquad$
$B$3$l$b(B$x$$B<4BP>N$@$+$i(B$0<x$$B$N>l9g$G9M$($l$PNI$$!#(B
\begin{equation}
\frac{dE}{dx}=-\frac{1}{4mx^3}(\hbar^2-4mCx^3)
\end{equation}
$B$H$J$k!#(B\\
$B=>$C$F0J2<$NA}8:I=$rF@$k!#(B\\

\begin{tabular}{c|rrrrr}
$x$&0&$\qquad$&$\sqrt[3]{\frac{\hbar^2}{4mC}}$&$\qquad$&$\infty$\\ \hline
$\frac{dE}{dx}$&$\qquad$&$-$&$0$&+&0\\  \hline
$E$&$\qquad$&$\searrow$&$\qquad$&$\nearrow$&+$\infty$\\
\end{tabular}\\
\\
$E$$B$,:G>.$H$J$k$N$O!"A}8:I=$h$j(B\\
$x=\displaystyle{\sqrt[3]{\frac{\hbar^2}{4mC}}}$$B$N$H$-$G!"$3$N$H$-(B
\begin{equation}
E=T+U=\frac{1}{2}C^{2/3}\sqrt[3]{\frac{\hbar^2}{4m}}+C^{2/3}\sqrt[3]{\frac{\hbar^2}{4m}}=\frac{3}{2}C^{2/3}\sqrt[3]{\frac{\hbar^2}{4m}}
\end{equation}
$B$^$?!"(B$T:U=1:2$$B$G$"$j!"$3$l$O%S%j%"%kDjM}$rK~$?$9!#(B
\item[(c)]
\begin{equation}
E=\frac{\hbar^2}{8mx^2}+x^4C
\end{equation}
$B$H$J$k!#(B\\
$B$3$l$b(B$x$$B<4BP>N$@$+$i(B$0<x$$B$N>l9g$G9M$($l$PNI$$!#(B
\begin{equation}
\frac{dE}{dx}=-\frac{1}{4mx^3}(\hbar^2-16mCx^6)
\end{equation}
$B$H$J$k!#(B\\
$B=>$C$F0J2<$NA}8:I=$rF@$k!#(B\\

\begin{tabular}{c|rrrrr}
$x$&0&$\qquad$&$\sqrt[6]{\frac{\hbar^2}{16mC}}$&$\qquad$&$\infty$\\ \hline
$\frac{dE}{dx}$&$\qquad$&$-$&$0$&+&0\\  \hline
$E$&$\qquad$&$\searrow$&$\qquad$&$\nearrow$&+$\infty$\\
\end{tabular}\\
\\
\\
$E$$B$,:G>.$H$J$k$N$O!"A}8:I=$h$j(B\\
$x=\displaystyle{\sqrt[6]{\frac{\hbar^2}{16mC}}}$$B$N$H$-$G!"$3$N$H$-(B
\begin{equation}
E=T+U=2C^{1/3}\left({\frac{\hbar^2}{16m}}\right)^{2/3}+C^{1/3}\left({\frac{\hbar^2}{16m}}\right)^{2/3}=3C^{1/3}\left({\frac{\hbar^2}{16m}}\right)^{2/3}
\end{equation}
$B$^$?!"(B$T:U=2:1=4:2$$B$G$"$j!"$3$l$O%S%j%"%kDjM}$rK~$?$9!#(B

\item[(d)]
$a\leq 0$$B$d(B$C\leq 0$$B$G$O4pDl>uBV$NI>2A$,$G$-$J$$$N$G(B$0<a$$B$+$D(B$0<C$$B$H$9$k!#(B
$BLdBjJ8$h$j(B
\begin{equation}
E=\frac{\hbar^2}{8mx^2}-e^{-ax^2}C\label{eq:Q5-5-1}
\end{equation}
$B$3$l$b(B$x$$B<4BP>N$@$+$i(B$0<x$$B$N>l9g$G9M$($l$P$h$$!#(B
\begin{eqnarray*}
E'&=&-\frac{\hbar^2}{4mx^3}+2aCxe^{-ax^2}\\
&=&2aCx\left(e^{-ax^2}-\frac{\hbar^2}{8maCx^4}\right)\\
\end{eqnarray*}
$f(x)$$B$G<!$N4X?t$rI=$9$3$H$K$9$k!#(B
\[ e^{-ax^2}-\frac{\hbar^2}{8maCx^4} \]
$B$3$l$O(B$0<x$$B$K$*$$$F(B$E'$$B$HId9f$,F1$8$GO"B3$J4X?t$G$"$k!#(B
\begin{displaymath}
f(x)\rightarrow\infty\;(x\rightarrow+0)
\end{displaymath}
\begin{displaymath}
f\left(\sqrt{\frac{1}{a}}\right)=\frac{1}{e}-\frac{a\hbar^2}{8mC}>0
\qquad(\frac{a\hbar^2}{8mC}\ll 1 $B$K$h$k(B)
\end{displaymath}
$B$K$h$C$F!"$"$k(B$0<x_0<\sqrt{1\left/a\right.}$$B$,B8:_$7$F(B$0=f(x_0)$$B$H$J$k!#(B
\[ f'=-2axe^{-ax^2}+\frac{\hbar^2}{2maCx^5} \]
$B$+$i!"(B
\[ 0=f(x_0)=e^{-a{x_0}^2}-\frac{\hbar^2}{8maC{x_0}^4}\Rightarrow
f'(x_0)=\frac{\hbar^2}{4mCx_0^3}\left(\frac{2}{ax_0^2}-1\right)>0 \]
$B$H$J$k$N$G!J(B$x_0<\sqrt{1\left/a\right.}\Rightarrow ax_0^2<1$$B$rMQ$$(B
$B$?!#!K!"(B$x_0$$B$,0l0U$KDj$^$j!"$=$N6aK5$G(B$f'$$B$,C1D4A}2C$7$F$$$k$3$H$b$o$+(B
$B$k!#$h$C$F(B$E'(x_0)=0$$B$G$"$j!"(B$x_0$$B$N6aK5$K$*$$$F$O!"(B
\[ x<x_0\Rightarrow E'(x)<0,\;x_0<x\Rightarrow 0<E'(x) \]
$B$H$J$k!#$h$C$F(B$E$$B$O(B$x_0$$B$K$*$$$F6K>.$H$J$j!"(B$x=x_0$$B$N$H$-$,4pDl>uBV$G$"(B
$B$k$3$H$,$o$+$C$?!#(B$a{x_0}^2<1\Rightarrow e^{-a{x_0}^2} \sim 1$$B$H!"(B
\begin{displaymath}
1\sim e^{-a{x_0}^2}=\frac{\hbar^2}{8maC{x_0}^4}\Rightarrow
{x_0}^2\sim\sqrt{\frac{\hbar^2}{8maC}}
\end{displaymath}
$B$r$D$+$C$F<0(B(\ref{eq:Q5-5-1})$B$+$i4pDl>uBV$N%(%M%k%.!<(B$E(x_0)$$B$NCM$O<!$N(B
$B$h$&$KI>2A$G$-$k!#(B
\[ E(x_0)\sim\sqrt{\frac{\hbar^2aC}{8m}}-C \]

\end{description}
}

