%$BNL;RJ*M}3X(B:$B1i=,LdBj(B5-3$B2rEz(B
%\documentclass{jsarticle}
%\begin{document}
%\title{$BNL;RO@652J=qLdBj>\2r!J3NG'MQ!K(B}
%\author{$B>.>>K~?N(B}
%$B=$@5<T(B: 240423G $B:4F#?.M5(B
%\date{2002.1.3}
%$B=$@5G/7nF|(B: /02/11/28
%$B=$@5<TEE;R%a!<%k(B: g240423@mail.ecc.u-tokyo.ac.jp 
%\maketitle

\mondai
{g140760 $B>.>>K~?N(B}
{2002/1/13}
{g140760@mail.ecc.u-tokyo.ac.jp}
{5-3}
{$ \varphi (x) = \pi ^{-1/4}\textit{e}^{-x^2/2} $$B$KBP$7!"652J=q(B(5.19),
(5.20)$B$N(B$\Delta x , \Delta p $$B$r7W;;$;$h!#(B}
{

$BLdBj$r2r$/$?$a$K!"<!$N@QJ,8x<0$r>ZL@$7$F$*$/!#(B
\begin{equation}
I(a)=\int^{\infty}_{-\infty}\textit{e}^{-ax^2}dx= \sqrt{\frac{\pi}{a}}
\label{eq:Q5-3-1}
\end{equation}
\
($B>ZL@!K(B
$B6v4X?t$G$"$k$3$H$J$I$rMQ$$$F(B,
\begin{equation}
I(a) = 2 \int^{\infty}_{0}\textit{e}^{-ax^2}dx \leq 2 \int^{\infty}_{0}x
\textit{e}^{-ax^2}dx \label{eq:Q5-3-2}
\end{equation}
$B:G1&JU$N@QJ,$O<}B+$9$k$N$G!"(B$I(a)$$B$b<}B+$9$k!#$=$3$G$^$:!"(B
\begin{equation}
I'(p) = \int^{p}_{0}\textit{e}^{-ax^2}dx \label{eq:Q5-3-3}
\end{equation}
$B$H$*$/!#1&2<?^$N@5J}7A$NNN0h$r(B$D_1$$B$H$9$l$P!"(B
$$ \{ I'(p) \}^2 = \biggl\{ \int^{p}_{0}\textit{e}^{-ax^2}dx
\biggr\} \biggl\{ \int^{p}_{0}\textit{e}^{-ay^2}dy \biggr\}
=$B!!(B\int \! \! \int_{D_1} \textit{e}^{-a(x^2+y^2)} dxdy =\int
\! \! \! \int_{D_1}r \textit{e}^{-ar^2}drd \theta \label{eq:Q5-3-4} $$
\begin{figure}[h]
\begin{flushright}
\begin{picture}(80,80)
\put(4,8){\vector(1,0){60}}
\put(4,8){\vector(0,1){60}}
\put(4,8){\framebox(40,40){}}
\put(4,8){\line(1,1){40}}
\put(-4,0){0}
\put(56,0){x}
\put(-4,64){y}
\put(-4,48){p}
\put(44,0){p}
\put(22,24){$\sqrt{2}$p}
\end{picture}
\end{flushright}
\end{figure}
$B@QJ,NN0h$NBg>.4X78$+$i<!$NITEy<0$,@.$jN)$D!#(B
$$ \int^{\frac{\pi}{2}}_{0}\int^{p}_{0}r \textit{e}^{-ar^2}drd\theta
\leq \{ I(p) \}^2 \leq \int^{\frac{\pi}{2}}_{0}\int^{\sqrt{2}p}_{0}r
\textit{e}^{-ar^2}drd\theta \label{eq:Q5-3-5} $$
$B:G:8JU$N@QJ,$O!"(B
\begin{equation}
\int^{\frac{\pi}{2}}_{0} \int^{p}_{0}r \textit{e}^{-ar^2}drd\theta =
\int^{\frac{\pi}{2}}_{0} \left[ - \frac{1}{2a} \textit{e}^{-ar^2}
\right] ^{p} _{0} d \theta = \frac{\pi}{4a}(1 - e^{-ap^2}) \label{eq:Q5-3-6}
\end{equation}
$B$3$N<0$N(Bp$B$r(B$\sqrt{2}p$$B$KCV$-49$($k$H!":G1&JU$N@QJ,$O(B
\begin{equation}
\int^{\frac{\pi}{2}}_{0} \int^{\sqrt{2}p}_{0}r
\textit{e}^{-ar^2}drd\theta = \frac{\pi}{4a}(1 - e^{-2ap^2})
\label{eq:Q5-3-7}
\end{equation}
(\ref{eq:Q5-3-6}),(\ref{eq:Q5-3-7})$B$O$I$A$i$b(B$ p \to \infty $
$B$H$9$k$H(B
$$\frac{\pi}{4a}$$
$B$K<}B+$9$k$N$G!"(B
\begin{equation}
\lim_{p \to \infty} I'(p) = \sqrt{\frac{\pi}{4a}} =
\frac{1}{2}\sqrt{\frac{\pi}{a}} \label{eq:Q5-3-8}
\end{equation}
(\ref{eq:Q5-3-2}),(\ref{eq:Q5-3-3}),(\ref{eq:Q5-3-8})$B$h$j!"(B
$$
I(a) = 2 \lim_{p \to \infty} I'(p) = \sqrt{\frac{\pi}{a}} \label{eq:Q5-3-9}
$$
\\

$B$3$l$rMQ$$$F$3$NLdBj$r2r$/!#(B
$$ \varphi (x) = \pi ^{-1 / 4}\textit{e}^{-x^2 / 2} \label{eq:Q5-3-10} $$
$B$h$j!"0LCV$N4|BTCM$O!"(B
$$ \bar{x} =\int^{\infty}_{-\infty} \varphi ^{*} x \varphi dx =
\frac{1}{\sqrt{\pi}}\int^{\infty}_{-\infty} x \textit{e}^{-x^2}dx =
\frac{1}{\sqrt{\pi}} \left[-\frac{1}{2}\textit{e}^{-x^2} \right]
^{\infty}_{-\infty} = 0 \label{eq:Q5-3-11} $$
$B$h$C$F0LCV$NIT3NDj$5$O(B
\begin{equation}
\Delta x = \biggl\{ \int^{\infty}_{-\infty} \varphi ^{*} (x - \bar{x})^2
\varphi dx \biggr\} ^{\frac{1}{2}} = \biggl\{ \int^{\infty}_{-\infty}
\varphi ^* x^2 \varphi dx \biggr\}^{\frac{1}{2}} = \biggl\{
\frac{1}{\sqrt{\pi}}\int^{\infty}_{-\infty} x^2 \textit{e}^{-x^2} dx
\biggr\}^{\frac{1}{2}} \label{eq:Q5-3-12}
\end{equation}
(\ref{eq:Q5-3-1})$B<0$HItJ,@QJ,$K$h$j!":G1&JU$N@QJ,$O(B
\begin{equation}
\int^{\infty}_{-\infty}x^2 \textit{e}^{-x^2} dx = \left[
-\frac{x}{2}\textit{e}^{-x^2} \right]^{\infty}_{-\infty} -
\int^{\infty}_{-\infty} \left( -\frac{1}{2}\textit{e}^{-x^2} dx \right)
= \frac{\sqrt{\pi}}{2} \label{eq:Q5-3-13}
\end{equation}
(\ref{eq:Q5-3-12})$B$KBeF~$7$F!"(B
\begin{equation}
\Delta x = \biggl\{ \frac{1}{\sqrt{\pi}}\frac{\sqrt{\pi}}{2}
\biggr\}^{\frac{1}{2}} = \frac{1}{\sqrt{2}} \label{eq:Q5-3-14}
\end{equation}
$B$rF@$k!#(B
$BA4$/F1MM$K$7$F!"1?F0NL$N4|BTCM$O(B
$$ \bar{p} = \int ^{\infty}_{-\infty} \varphi ^{*}
\left(\frac{\hbar}{\textit{i}} \frac{d}{dx} \right) \varphi dx =
\frac{\textit{i}\hbar}{\sqrt{\pi}}\int^{\infty}_{-\infty}x
\textit{e}^{-x^2}dx = 0 \label{eq:Q5-3-15}$$
$B$h$C$F1?F0NL$NIT3NDj$5(B
\begin{eqnarray}
&\Delta p = \biggl\{ \int^{\infty}_{-\infty} \varphi ^* \left(
\frac{\hbar}{\textit{i}}\frac{d}{dx} \right)^{2} \varphi dx \biggr\}
^{\frac{1}{2}} \\
& \qquad\qquad\quad= \biggl\{ -\frac{\hbar^2}{\sqrt{\pi}}
\int^{\infty}_{-\infty} \textit{e}^{-x^2 / 2}\frac{d}{dx}\left( -x
\textit{e}^{-x^2 / 2 } \right) dx \biggr\}^{\frac{1}{2}} \\
&= \biggl\{ -\frac{\hbar^2}{\sqrt{\pi}} \int^{\infty}_{-\infty} (x^2 -
1)\textit{e}^{-x^2} dx \biggr\} ^{\frac{1}{2}} \label{eq:Q5-3-16}
\end{eqnarray}
$B:G1&JU$N@QJ,$O(B(\ref{eq:Q5-3-1}),(\ref{eq:Q5-3-13})$B<0$h$j!"(B
\begin{equation}
\int^{\infty}_{-\infty} (x^2 -1)\textit{e}^{-x^2} dx =
\frac{\sqrt{\pi}}{2} - \sqrt{\pi} = -\frac{\sqrt{\pi}}{2} \label{eq:Q5-3-17}
\end{equation}
$B$3$l$r(B(\ref{eq:Q5-3-16})$B<0$KBeF~$7$F!"(B
\begin{equation}
\Delta p = \biggl\{ -\frac{\hbar^2}{\sqrt{\pi}} \left(
-\frac{\sqrt{\pi}}{2} \right) \biggr\}^{\frac{1}{2}} =
\frac{\hbar}{\sqrt{2}} \label{eq:Q5-3-18}
\end{equation}
$B$,5a$^$k!#(B(\ref{eq:Q5-3-14}),(\ref{eq:Q5-3-18})$B<0$OIT3NDj@-86M}$NITEy<0(B
$$
\Delta x \cdot \Delta p \geq \frac{\hbar}{2}
$$
$B$rK~B-$7$F$$$k!#!J2rEz=*$o$j!K(B}

%\end{document}




