量子力学Ⅰ/水素原子/メモ のバックアップソース(No.1)

更新 
[[量子力学Ⅰ/水素原子]]

* 解く [#k100457f]

　&math(\frac{\PD^2\chi}{\PD\rho^2}+\left\{\frac{2}{\rho}-\frac{l(l+1)}{\rho^2}\right\}\chi-\frac{1}{n^2}\chi=0);

&math(\chi=X(\rho)e^{-\rho/n}); と置けば、

　&math(
\frac{\PD^2}{\PD\rho^2}X(\rho)e^{-\rho/n}
&=\frac{\PD}{\PD\rho}X'(\rho)e^{-\rho/n}-\frac{1}{n}\frac{\PD}{\PD\rho}X(\rho)e^{-\rho/n}\\
&=X''(\rho)e^{-\rho/n}-\frac{2}{n}X'(\rho)e^{-\rho/n}+\frac{n}{n^2}X(\rho)e^{-\rho/n}\\
);

を使って、

　&math(
X''(\rho)-\frac{2}{n}X'(\rho)+\cancel{\frac{1}{n^2}X(\rho)}
+\left\{\frac{2}{\rho}-\frac{l(l+1)}{\rho^2}\right\}X(\rho)-\cancel{\frac{1}{n^2} X(\rho)}=0
);

　&math(
X''(\rho)-\frac{2}{n}X'(\rho)+\left\{\frac{2}{\rho}-\frac{l(l+1)}{\rho^2}\right\}X(\rho)=0
);

&math(X(\rho)=\sum_{i=0}^\infty c_k\rho^k); と置けば、

　&math(
\sum_{i=0}^\infty k(k-1)c_k\rho^{k-2}-\frac{2}{n}\sum_{i=0}^\infty kc_k\rho^{k-1}+\left\{\frac{2}{\rho}-\frac{l(l+1)}{\rho^2}\right\}\sum_{i=0}^\infty c_k\rho^k=0
);

　&math(
\sum_{i=0}^\infty (k+2)(k+1)c_{k+2}\rho^k-\frac{2}{n}\sum_{i=0}^\infty (k+1)c_{k+1}\rho^k+\sum_{i=0}^\infty 2c_k\rho^{k-2}-\sum_{i=0}^\infty l(l+1)c_k\rho^{k-1}=0
);

　&math(
\sum_{i=0}^\infty (k+2)(k+1)c_{k+2}\rho^k-\frac{2}{n}\sum_{i=0}^\infty (k+1)c_{k+1}\rho^k+\sum_{i=-2}^\infty 2c_{k+2}\rho^k-\sum_{i=-1}^\infty l(l+1)c_{k+1}\rho^{k}=0
);

　&math(
&2c_0\rho^{-2}+\{2c_{1}-l(l+1)c_0\}\rho^{-1}+\\
&\sum_{i=0}^\infty \big[\big\{(k+2)(k+1)+2\big\}c_{k+2}-\big\{2(k+1)/n+l(l+1)\big\}c_{k+1}\big]\rho^k=0\\
);

すなわち、

　&math(
&c_0=0\\
&c_1=\frac{l(l+1)}{2}c_0\\
&\big\{(k+2)(k+1)+2\big\}c_{k+2}-\big\{2(k+1)/n+l(l+1)\big\}c_{k+1}=0
);
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