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% $B:n(B  $B@.(B  $B<T(B : g240947 $B0f>eF;M:(B
% $B:n@.G/7nF|(B : /02/12/19
% $BEE;R%a!<%k(B : micchan@bf7.so-net.ne.jp
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%  \mondai{$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
{g240947 $B0f>eF;M:(B}
{02/12/19}
{micchan@bf7.so.net.ne.jp}
{6-3}
{$BNcBj(B6.1$B$G(B$A$$B$N8GM-CM$,=L=E$7$F$$$k>l9g$K$b!"(B
$\varphi$$B$,(B$B$$B$N8GM-4X?t$G$"$k$3$H$r<($;!#(B}
{
$A$$B$N8GM-CM$,(B$n$$B=E$K=L=E$7$F$$$l$P!"(B
$A$$B$NFHN)$J8GM-4X?t$,(B$n$$B8D$"$k!#(B
$B$3$l$r(B$\varphi_{k}$ $\left(k=1,2,\dots,n\right)$$B$H$*$/!#(B
$\varphi_{k}$$B$r@55,D>8r2=$7$?$b$N(B$<\varphi_{k}|\varphi_{k^{'}}> =
\delta_{kk^{'}}$
$B$H$7$F$b!"0lHL@-$r<:$o$J$$!#(B
$B$3$N$H$-!"(B$\varphi$$B$r8GM-4X?t(B$\varphi_{k}$$B$N0l<!7k9g!"(B
\begin{eqnarray}
\varphi &=& \sum_{k=1}^n c_k \varphi_k , \label{eq:Q6-3-2}
\end{eqnarray}
$B$H$*$1$k!#$3$3$G!"(B$c_k$$B$r?7$7$$4X?t(B$\varphi$$B$,1i;;;R(B$B$$B$N8GM-CM(B$b$$B$KBP$9$k8G(B
$BM-4X?t(B
\begin{eqnarray}
B\varphi &=& b\varphi ,  \label{eq:Q6-3-3}
\end{eqnarray}
$B$G$"$k$h$&$KA*$Y$P$h$$!#<0(B(\ref{eq:Q6-3-3})$B$K<0(B(\ref{eq:Q6-3-2})$B$rBeF~$9$k!#(B
\begin{eqnarray}
B\sum_{k=1}^n c_k \varphi_k &=& b\sum_{k=1}^n c_k \varphi_k .
\end{eqnarray}
$B$3$3$G!"N>JU$K:8$+$i(B$\varphi_{k^{'}}^\ast$$B$r3]$1(B$r$$B$K$D$$$F@QJ,$9$k!#Ey<0$N(B
$B:8JU$N@QJ,(B
\begin{eqnarray}
\int\varphi_{k^{'}}^{\ast}B\varphi_{k}dr ,
\end{eqnarray}
$B$O!"BP1~$9$k9TNsMWAG(B$b_{k^{'}k}$$B$KEy$7$$!#1&JU$G$O(B$k \neq k^{'}$$B$N$H$-!"@QJ,(B
\begin{eqnarray}
\int\varphi_{k^{'}}^{\ast}\varphi_{k}dr ,
\end{eqnarray}
$B$OD>8r4X78$K$h$C$F$9$Y$F>C$(!"$^$?5,3J2=>r7o$K$h$C$F(B$k=k^{'}$$B$N$H$-!"(B
\begin{eqnarray}
\int\varphi_{k}^{\ast}\varphi_{k}dr &=&1 ,
\end{eqnarray}
$B$H$J$k!#$7$?$,$C$F!"(B
\begin{eqnarray}
\sum_{k=1}^n b_{k^{'}k}c_{k} &=& bc_{k^{'}}  \left(k^{'}=1,2,\dots,n\right),
\label{eq:Q6-3-4}
\end{eqnarray}
$B$rF@$k!#JQ7A$9$k$H(B
\begin{eqnarray}
\sum_{k=1}^n(b_{k^{'}k}-b\delta_{k^{'}k})c_{k} &=& 0 ,
\end{eqnarray}
$B$3$l$,78?t(B$c_{k}$$B$r7hDj$9$kF1<!O"N)Be?tJ}Dx<0$G$"$k!#(B
$B$3$NJ}Dx<0$,(B$0$$B0J30$N2r$r;}$D$N$O!"(B
$BJ}Dx<0$N78?t$G:n$C$?9TNs<0$,(B$0$$B$N>l9g!"$9$J$o$A!"(B
\begin{eqnarray}
\det
\left|
\begin{array}{cccc}
b_{11}-b & b_{12} & \dots & b_{1n} \\
b_{21} & b_{22} - b & \dots & b_{2n} \\
\dots \\
b_{n1} & b_{n2} & \dots & b_{nn} - b
\end{array}
\right|
&=&0,  \label{eq:Q6-3-5}
\end{eqnarray}
$B$N>l9g$K8B$i$l$k!#$3$N<0$O(B$b$$B$N(B$n$$B<!J}Dx<0$J$N$G!"(B
$B:,(B$b_1,b_2,\dots,b_n$$B$,5a$a$i$l$k!#(B
$B$*$N$*$N$N:,(B$b_\alpha$$B$KBP1~$7$F!"<0(B(\ref{eq:Q6-3-4})$B$N2r!"(B
$c_{\alpha 1},c_{\alpha 2},\dots,c_{\alpha n}$$B$,F@$i$l!"(B
$B$7$?$,$C$F!"<0(B(\ref{eq:Q6-3-2})$B$+$i!"4X?t(B
\begin{eqnarray}
\varphi_{\alpha} &=& \sum_{k=1}^n c_{\alpha k}\varphi_{k}, \label{eq:Q6-3-6}
\end{eqnarray}
$B$,Dj$a$i$l$k!#?7$7$$4X?t(B$\varphi_{\alpha} $$ \left(\alpha =
1,2,\dots,n\right) $
$B$O!"(B$\varphi_{k}$$B$N0l<!7k9g$G$"$k$+$i!"(B
$B1i;;;R(B$B$$B$N8GM-CM(B$b$$B$KBP$9$k8GM-4X?t$G$"$j!"(B
$BF1;~$K1i;;;R(B$A$$B$N8GM-CM(B$a = a_{1},a_{2},\dots,a_{n}$
$B$K$=$l$>$lBP$9$k8GM-4X?t$G$b$"$k!#$h$C$F!"(B$A$$B$N8GM-CM$,=L=E$9$k>l9g$G$b!"(B
$[A,B] = 0$$B$J$i!"(B$A$$B$H(B$B$$B$OF1;~8GM-CM$r;}$D$3$H$,<($5$l$?!#(B}
