%<PLAINTEXT>
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% $BNL;RJ*M}3X(B : $B1i=,LdBj(B 6-5 $B2rEz(B
% $B:n(B  $B@.(B  $B<T(B : 240897H $B5Z@n0l@?(B
% $B:n@.G/7nF|(B : 2002/10/31
% $BEE;R%a!<%k(B : g240897@mail.ecc.u-tokyo.ac.jp
% 
\mondai{240897H $B5Z@n0l@?(B}
{2002/10/31}
{g240897@mail.ecc.u-tokyo.ac.jp}
{6-5}
{
($BGHB+$N;~4VH/E8(B)$BGHF04X?t(B$\varphi$$B$,8GM-4X?t$G$J$$>l9g$K$O!"GHF04X?t$O;~4V$K0MB8$9$k%7%e%l%G%#%s%,!<J}Dx<0(B(6.33)$B$K=>$C$F1?F0$9$k!#<+M3N3;R(B($\displaystyle{{\cal H}=\frac{p^2}{2m}}$)$B$N>l9g!"GHF04X?t$N;~4VH/E8(B(6.33)$B$O!"(B
\begin{eqnarray}
\qquad \qquad \qquad \qquad \qquad  i\hbar\frac{\partial}{\partial t}\varphi = -\frac{\hbar ^2}{2m}\frac{\partial ^2}{\partial x^2}\varphi \qquad \qquad \qquad \qquad (6.39) \nonumber
\end{eqnarray}
$B$K$J$k!#(B(6.39)$B$r0J2<$NJ}K!$rMQ$$$F2r$/!#3FLd$KEz$($h!#(B
\\
\begin{description}
\item[(a)]
$\varphi(x) $B$N(Bx$$B$K4X$9$k%U!<%j%(JQ49$r(B
\begin{eqnarray*}
f(k)=\int_{-\infty}^{\infty}e^{ikx}\varphi(x)dx
\end{eqnarray*}
$B$H$*$/$H(B(6.39)$B$O!"(B
\begin{eqnarray*}
\qquad \qquad \qquad \qquad i\hbar\frac{\partial}{\partial t}f(k) = \frac{\hbar^2k^2}{2m}f(k) \qquad \qquad \qquad \quad (6.40)
\end{eqnarray*}
$B$H$J$k$3$H$r<($;!#(B\\

\item[(b)]
(6.40)$B$N2r$O!"(B
\begin{eqnarray*}
\qquad \qquad \qquad \quad f(k)=f(k)|_{t=0}e^{-i\hbar k^2t/2m} \qquad \qquad \qquad (6.41)
\end{eqnarray*}
$B$G$"$k$3$H$r<($;!#(B$f(k)|_{t=0}$B$O(Bt=0$B$G$N(Bf(k)$B$NCM$G$"$k!#(B$\\
\item[(c)]
$B$h$C$F5a$a$k;~4V$K0MB8$9$kGHF04X?t$O!"(B(6.41)$B$r%U!<%j%(5UJQ49(B
\begin{eqnarray*}
\qquad \qquad \qquad \varphi(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty}f(k)e^{-ikx}dk \qquad \qquad \qquad (6.42)
\end{eqnarray*}
$B$9$k$3$H$GF@$i$l$k!#(B$t=0$$B$N$H$-!"GHF04X?t$,(B
\begin{eqnarray*}
\varphi(x) = \frac{1}{\sqrt{\pi}a}e^{-x^2/a^2} 
\end{eqnarray*}
$B$N$H$-;~4V(B$t$$B$G$N(B$\varphi(x,t)$$B$r5a$a$h!#(B

\end{description}
}
{
\begin{description}
\item[(a)]
\begin{eqnarray*}
i\hbar\frac{\partial}{\partial t}f(k) &=& i\hbar\frac{\partial}{\partial t}\int_{-\infty}^{\infty}e^{ikx}\varphi(x)dx \\
 &=& i\hbar\int_{-\infty}^{\infty}e^{ikx}\frac{\partial}{\partial t}\varphi(x)dx \\
 &=& -\frac{\hbar^2}{2m}\int_{-\infty}^{\infty}e^{ikx}\frac{\partial^2}{\partial x^2}\varphi(x)dx \qquad ((6.39)$B$h$j(B) \\
&=& -\frac{i\hbar^2k}{2m}\int_{-\infty}^{\infty}e^{ikx}\frac{\partial}{\partial x}\varphi(x)dx \\
&=& \frac{\hbar^2k^2}{2m}\int_{-\infty}^{\infty}e^{ikx}\varphi(x)dx \qquad (2$B2sItJ,@QJ,$r9T$C$?(B) \\
&=& \frac{\hbar^2k^2}{2m}f(k)
\end{eqnarray*}
$B$h$C$F(B(6.40)$B$,<($;$?!#(B
\item[(b)]
(6.40)$B$h$j(B
\begin{eqnarray*}
\frac{\partial f(k)}{\partial t}\frac{1}{f(k)} &=& -\frac{i\hbar k^2}{2m} \\
 \log|f(k)| &=& -\frac{i\hbar k^2t}{2m}+C($B@QJ,Dj?t(B) \\
 f(k) &=& e^{-i\hbar k^2t/2m+C} 
\end{eqnarray*}
$t=0$B$N$H$-(B$
\begin{eqnarray*}
f(k)|_{t=0} &=& e^C 
\end{eqnarray*}
$B$G$"$k$+$i!"(B
\begin{eqnarray*}
 f(k)=f(k)|_{t=0}e^{-i\hbar k^2t/2m}
\end{eqnarray*}
$B$H$J$k!#(B
\item[(c)]
$B$^$:!"(B$f(k)|_{t=0}$$B$r5a$a$k!#(B
\begin{eqnarray*}
f(k)|_{t=0} &=& \int_{-\infty}^{\infty}e^{ikx}\varphi(x,0)dx \\
            &=& \frac{1}{\sqrt{\pi}a}\int_{-\infty}^{\infty}e^{-x^2/a^2 \, + \, ikx}dx \\
            &=& \frac{1}{\sqrt{\pi}a}\int_{-\infty}^{\infty}e^{-(x\, -\, ika^2/2)^2/a^2 \, - \, k^2a^2/4}dx
\end{eqnarray*}
$B$3$3$G!"(B$\displaystyle{\int_{-\infty}^{\infty}e^{-x^2}dx = \sqrt{\pi}}$$B$h$j(B

\begin{eqnarray*}
\int_{-\infty}^{\infty}e^{-(x\, -\, ika^2/2)^2/a^2}dx=\sqrt{\pi}a
\end{eqnarray*}
$B$G$"$k$+$i!"(B
\begin{eqnarray*}
\frac{1}{\sqrt{\pi}a}\int_{-\infty}^{\infty}e^{-(x\, -\, ika^2/2)^2/a^2 \, - \, k^2a^2/4}dx &=& e^{-k^2a^2/4} 
\end{eqnarray*}
$B=>$C$F!"(B
\begin{eqnarray*}
f(k)|_{t=0}&=& e^{-k^2a^2/4} 
\end{eqnarray*}
$B$G$"$k!#$9$k$H(B(6.42)$B$O(B
\begin{eqnarray*}
\varphi(x) &=& \frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-(a^2/4\, +\, i\hbar t/2m)k^2-ikx}dk \\
 &=& \frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-bk^2-ikx}dk \qquad \qquad \left( b=\frac{a^2}{4} + \frac{i\hbar t}{2m} $B$H$*$$$?(B\right) \\
 &=& \frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-b(k\, +\, ix/2b)^2-x^2/4b^2}dk \\
 &=& \frac{1}{2\sqrt{\pi b}}e^{-x^2/4b^2} \qquad \qquad \left( \int_{-\infty}^{\infty}e^{-b(k\, +\, ix/2b)^2} = \sqrt{\frac{\pi}{b}}$B$h$j(B\right) 
\end{eqnarray*}
$B$H$J$k!#(B
\end{description}
}
