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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 /02/11/26
% $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}
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\mondai
{241025B $B;32<!!??(B}
{/02/11/18}
{g241025@mail.ecc.u-tokyo.ac.jp}
{5-4}
{
%----$BLdBjJ8(B----%

$B!JD4OB?6F0;R!KD4OB?6F0;R$N%O%_%k%H%K%"%s(B(5.37)$B$K$*$$$F!"(B$\alpha=m\
omega_0/\hbar$$B$H$*$/$H!"%O%_%k%H%K%"%s$O(B
\begin{eqnarray}
\left(\frac{d^2}{dx^2}-\alpha^2
x^2\right)\varphi=-\frac{2mE}{\hbar^2}\varphi\nonumber  \qquad \qquad
\qquad \qquad(5.42)
\end{eqnarray}
$B$K$J$k!#$3$N$H$-!"(B$\varphi_n(\xi)=H_n(\xi)e^{-\xi^2/2},
(\xi=\sqrt{\alpha}x)$B$O!"(B(5.42)$B$N8GM-4X?t$G$"$k$3$H$r<($;!#$3$3$G%(%k%_!<(B
$B%HB?9`<0(BH_n(\xi)$
\begin{eqnarray}
H_n(\xi)=(-1)^ne^{\xi^2}\frac{d^n}{d\xi^n}e^{-\xi^2}\nonumber  \qquad
\qquad \qquad \qquad(5.43)
\end{eqnarray}
$B$KBP1~$9$k8GM-CM$,(B$\left(n+\frac{1}{2}\right)\hbar\omega_0$$B$GM?$($i$l$k(B
$B$3$H$r<($;!#(B$H_n(\xi)$$B$N4X?t$N?6$kIq$$$rCN$k$?$a$K!"(B$H_n(\xi)$$B$N4X?t7A$r(B
$n=0,1,2$$B$G5a$a$F$_$h!#(B
}
{
$\xi=\sqrt{\alpha}x$B$h$j(B(5.42)$B$r(Bx$B$+$i(B\xi$$B$KJQ?tJQ49$9$k$H!"HyJ,J}Dx<0$O(B
\begin{displaymath}
\left(\alpha\frac{d^2}{d\xi^2}-\alpha\xi^2\right)\varphi(\xi)=-\frac{2mE}{\hbar^2}\varphi(\xi)\Longleftrightarrow
\left(\frac{d^2}{d\xi^2}-\xi^2\right)\varphi(\xi)=-\frac{2E}{\hbar\omega_0}\varphi(\xi)
\end{displaymath}
$B$K$J$k!#(B
\begin{displaymath}
\frac{d^2}{d\xi^2}\varphi_n(\xi)=H_n''e^{-\xi^2/2}-2\xi
H_n'e^{-\xi^2/2}+\xi^2H_ne^{-\xi^2/2}-H_ne^{-\xi^2/2}
\end{displaymath}
$B$h$j(B,
\begin{equation}
\frac{d^2}{d\xi^2}\varphi_n(\xi)-\xi^2\varphi_n(\xi)=\left(H_n''-2\xi
H_n'-H_n\right)e^{-\xi^2/2}\label{eq:Q-5-4-1}
\end{equation}
$B$H$J$k!#$3$3$G%(%k%_!<%HB?9`<0$NDj5A<0(B(5.43)$B$h$j!"(B
\begin{displaymath}
H_n'(\xi)=(-1)^n\left[2\xi
e^{\xi^2}\frac{d^n}{d\xi^n}e^{-\xi^2}+e^{\xi^2}\frac{d^{n+1}}{d\xi^{n+1}}e^{-\xi^2}\right],
\end{displaymath}
\begin{displaymath}
H_n''(\xi)=(-1)^n\left\{2e^{\xi^2}\frac{d^n}{d\xi^n}e^{-\xi^2}+(2\xi)^2
e^{\xi^2}\frac{d^n}{d\xi^n}e^{-\xi^2}+4\xi
e^{\xi^2}\frac{d^{n+1}}{d\xi^{n+1}}e^{-\xi^2}+e^{\xi^2}\frac{d^{n+2}}{d\xi^{n+2}}e^{-\xi^2}\right\},
\end{displaymath}
\begin{displaymath}
H_n''(\xi)-2\xi
H_n'(\xi)=(-1)^ne^{\xi^2}\left(2\frac{d^n}{d\xi^n}e^{-\xi^2}+2\xi\frac{d^{n+1}}{d\xi^{n+1}}e^{-\xi^2}+\frac{d^{n+2}}{d\xi^{n+2}}e^{-\xi^2}\right)
\end{displaymath}
$B$K$h$C$F!"(B
\begin{equation}
\frac{H_n''(\xi)-2\xi
H_n'(\xi)}{H_n(\xi)}=2+\left\{\left(2\xi\frac{d^{n+1}}{d\xi^{n+1}}e^{-\xi^2}+\frac{d^{n+2}}{d\xi^{n+2}}e^{-\xi^2}\right)\left/\frac{d^n}{d\xi^n}e^{-\xi^2}\right.\right\}\label{eq:Q-5-4-2}
\end{equation}
$B$3$3$G(B
\begin{displaymath}
(fg)^{\left(n\right)}=\sum^n_{i=0}{}_n C _i
f^{\left(i\right)}g^{\left(n-i\right)}
\end{displaymath}
$B$,(B$n$$B2sHyJ,2DG=$J0lHL$N4X?t(B$f, g$$B$K$D$$$F@.N)$7(B
\begin{displaymath}
\frac{d^{n+2}}{d\xi^{n+2}}
e^{-\xi^2}=\frac{d^{n+1}}{d\xi^{n+1}}\left(-2\xi e^{-\xi^2}\right)
\end{displaymath}
$B$G$"$k$+$i!"(B$f=-2\xi, g=e^{-\xi^2}$$B$H$*$1$P(B$f'=-2, f^{\left(j\right)}=0,
(2\leq j)$$B$h$j(B
\begin{displaymath}
\frac{d^{n+2}}{d\xi^{n+2}}
e^{-\xi^2}=-2\xi\frac{d^{n+1}}{d\xi^{n+1}}e^{-\xi^2}+\left(n+1\right)\left(-2\frac{d^n}{d\xi^n}e^{-\xi^2}\right)
\end{displaymath}
(\ref{eq:Q-5-4-1}),~(\ref{eq:Q-5-4-2})$B$H$"$o$;$F(B
\begin{displaymath}
\frac{d^2}{d\xi^2}\varphi_n(\xi)-\xi^2\varphi_n(\xi)=(-2n-1)\varphi_n(\xi)
\end{displaymath}
$B8GM-%(%M%k%.!<$O(B$-2(n+\frac{1}{2})=-2E_n/\hbar\omega_0\Longrightarrow
E_n=(n+\frac{1}{2})\hbar\omega_0$$B$H$J$C$?!#(B\\
\\
$H_n$$B$N6qBNE*$J7A$O(B$H_0(\xi)=1,H_1(\xi)=2\xi,H_2(\xi)=4\xi^2-2$$B$J$I$H$J(B
$B$k!#4JC1$J5"G<K!$K$h$C$F(B$\xi^n$$B$N78?t$,(B$2^n$$B$G!"$=$NB>$N9`$N78?t$,$9$Y$F(B
$B@0?t$G$"$k$3$H$,$o$+$k!#(B
}

