中心力場内の粒子の運動/メモ のバックアップソース(No.1)

更新

[[量子力学Ⅰ/中心力場内の粒子]]

* 球座標表示のラプラシアン [#o621468a]

&math(
\frac{\PD^2}{\PD x^2}
&=\Big(\sin\theta\cos\phi \frac{\PD}{\PD r}
+\frac{1}{r}\cos\theta\cos\phi \frac{\PD}{\PD \theta}
-\frac{\sin\phi}{r\sin\theta} \frac{\PD}{\PD \phi}\Big)^2\\

&=\sin\theta\cos\phi \frac{\PD}{\PD r}\Big(\sin\theta\cos\phi \frac{\PD}{\PD r}
+\frac{1}{r}\cos\theta\cos\phi \frac{\PD}{\PD \theta}
-\frac{\sin\phi}{r\sin\theta} \frac{\PD}{\PD \phi}\Big)\\
&\ \ \ +\frac{1}{r}\cos\theta\cos\phi \frac{\PD}{\PD \theta}
\Big(\sin\theta\cos\phi \frac{\PD}{\PD r}
+\frac{1}{r}\cos\theta\cos\phi \frac{\PD}{\PD \theta}
-\frac{\sin\phi}{r\sin\theta} \frac{\PD}{\PD \phi}\Big)\\
&\ \ \ -\frac{\sin\phi}{r\sin\theta} \frac{\PD}{\PD \phi}
\Big(\sin\theta\cos\phi \frac{\PD}{\PD r}
+\frac{1}{r}\cos\theta\cos\phi \frac{\PD}{\PD \theta}
-\frac{\sin\phi}{r\sin\theta} \frac{\PD}{\PD \phi}\Big)\\

&=\sin^2\theta\cos^2\phi \frac{\PD^2}{\PD r^2}
-\frac{\sin\theta\cos\theta\cos^2\phi}{r^2} \frac{\PD}{\PD \theta}
+\frac{\sin\theta\cos\theta\cos^2\phi}{r} \frac{\PD}{\PD r}\frac{\PD}{\PD \theta}
+\frac{\sin\phi\cos\phi}{r^2} \frac{\PD}{\PD \phi}
-\frac{\sin\phi\cos\phi}{r} \frac{\PD}{\PD r}\frac{\PD}{\PD \phi}\\
&\ \ \ 
+\frac{\cos^2\theta\cos^2\phi}{r} \frac{\PD}{\PD r}
+\frac{\sin\theta\cos\theta\cos^2\phi}{r} \frac{\PD}{\PD r}\frac{\PD}{\PD \theta}
+\frac{-\sin\theta\cos\theta\cos^2\phi}{r^2}\frac{\PD}{\PD \theta}
+\frac{\cos^2\theta\cos^2\phi}{r^2} \frac{\PD^2}{\PD \theta^2}\\
&\hspace{9cm}+\frac{\cos^2\theta\sin\phi\cos\phi}{r^2\sin^2\theta} \frac{\PD}{\PD \phi}
-\frac{\cos\theta\sin\phi\cos\phi}{r^2\sin\theta} \frac{\PD}{\PD \theta}\frac{\PD}{\PD \phi}
\\
&\ \ \ 
+\frac{\sin^2\phi}{r} \frac{\PD}{\PD r}
-\frac{\sin\phi\cos\phi}{r} \frac{\PD}{\PD r}\frac{\PD}{\PD \phi}
+\frac{\cos\theta\sin^2\phi}{r^2\sin\theta} \frac{\PD}{\PD \theta}
-\frac{\cos\theta\sin\phi\cos\phi}{r^2\sin\theta} \frac{\PD}{\PD \theta}\frac{\PD}{\PD \phi}
+\frac{\sin\phi\cos\phi}{r^2\sin^2\theta} \frac{\PD}{\PD \phi}
+\frac{\sin^2\phi}{r^2\sin^2\theta} \frac{\PD^2}{\PD \phi^2}\\

&=\sin^2\theta\cos^2\phi \frac{\PD^2}{\PD r^2}
-\frac{\sin\theta\cos\theta\cos^2\phi}{r^2} \frac{\PD}{\PD \theta}
+\frac{2\sin\theta\cos\theta\cos^2\phi}{r} \frac{\PD}{\PD r}\frac{\PD}{\PD \theta}
+\frac{\sin\phi\cos\phi}{r^2} \frac{\PD}{\PD \phi}
-\frac{2\sin\phi\cos\phi}{r} \frac{\PD}{\PD r}\frac{\PD}{\PD \phi}\\
&\ \ \ 
+\frac{\cos^2\theta\cos^2\phi}{r} \frac{\PD}{\PD r}
+\frac{-\sin\theta\cos\theta\cos^2\phi}{r^2}\frac{\PD}{\PD \theta}
+\frac{\cos^2\theta\cos^2\phi}{r^2} \frac{\PD^2}{\PD \theta^2}
+\frac{\cos^2\theta\sin\phi\cos\phi}{r^2\sin^2\theta} \frac{\PD}{\PD \phi}\\
&\ \ \ 
-\frac{2\cos\theta\sin\phi\cos\phi}{r^2\sin\theta} \frac{\PD}{\PD \theta}\frac{\PD}{\PD \phi}
+\frac{\sin^2\phi}{r} \frac{\PD}{\PD r}
+\frac{\cos\theta\sin^2\phi}{r^2\sin\theta} \frac{\PD}{\PD \theta}
+\frac{\sin\phi\cos\phi}{r^2\sin^2\theta} \frac{\PD}{\PD \phi}
+\frac{\sin^2\phi}{r^2\sin^2\theta} \frac{\PD^2}{\PD \phi^2}\\
);

&math(
\frac{\PD^2}{\PD y^2}
&=\Big(
\sin\theta\sin\phi \frac{\PD}{\PD r}
+\frac{1}{r}\cos\theta\sin\phi \frac{\PD}{\PD \theta}
+\frac{\cos\phi}{r\sin\theta} \frac{\PD}{\PD \phi}
\Big)^2\\
&=
\sin^2\theta\sin^2\phi \frac{\PD^2}{\PD r^2}
-\frac{\sin\theta\cos\theta\sin^2\phi}{r^2} \frac{\PD}{\PD \theta}
+\frac{2\sin\theta\cos\theta\sin^2\phi}{r} \frac{\PD}{\PD r}\frac{\PD}{\PD \theta}
-\frac{\sin\phi\cos\phi}{r^2} \frac{\PD}{\PD \phi}
+\frac{2\sin\phi\cos\phi}{r} \frac{\PD}{\PD r}\frac{\PD}{\PD \phi}\\
&\ \ \ 
+\frac{\cos^2\theta\sin^2\phi}{r} \frac{\PD}{\PD r}
+\frac{-\sin\theta\cos\theta\sin^2\phi}{r^2} \frac{\PD}{\PD \theta}
+\frac{\cos^2\theta\sin^2\phi}{r^2} \frac{\PD^2}{\PD \theta^2}
-\frac{\cos^2\theta\sin\phi\cos\phi}{r^2\sin^2\theta} \frac{\PD}{\PD \phi}
+\frac{2\cos\theta\sin\phi\cos\phi}{r^2\sin\theta} \frac{\PD}{\PD \theta}\frac{\PD}{\PD \phi}\\
&\ \ \ 
+\frac{\cos^2\phi}{r} \frac{\PD}{\PD r}
+\frac{\cos\theta\cos^2\phi}{r^2\sin\theta} \frac{\PD}{\PD \theta}
+\frac{-\sin\phi\cos\phi}{r^2\sin^2\theta} \frac{\PD}{\PD \phi}
+\frac{\cos^2\phi}{r^2\sin^2\theta} \frac{\PD^2}{\PD \phi^2}
);

&math(
\frac{\PD^2}{\PD z^2}
&=\Big(
\cos\theta \frac{\PD}{\PD r}
-\frac{1}{r}\sin\theta \frac{\PD}{\PD \theta}
\Big)^2\\
&=
\cos^2\theta \frac{\PD^2}{\PD r^2}
+\frac{\sin\theta\cos\theta}{r^2} \frac{\PD}{\PD \theta}
-\frac{2\sin\theta\cos\theta}{r} \frac{\PD}{\PD r}\frac{\PD}{\PD \theta}
\\
&\ \ \ 
+\frac{\sin^2\theta}{r} \frac{\PD}{\PD r}
+\frac{\sin\theta\cos\theta}{r^2} \frac{\PD}{\PD \theta}
+\frac{\sin^2\theta}{r^2} \frac{\PD^2}{\PD \theta^2}
);

足せばいい。

&math(
&\frac{\PD^2}{\PD x^2}+\frac{\PD^2}{\PD y^2}+\frac{\PD^2}{\PD z^2}\\
&=
\sin^2\theta\cos^2\phi \frac{\PD^2}{\PD r^2}
-\frac{\sin\theta\cos\theta\cos^2\phi}{r^2} \frac{\PD}{\PD \theta}
+\frac{2\sin\theta\cos\theta\cos^2\phi}{r} \frac{\PD}{\PD r}\frac{\PD}{\PD \theta}
+\frac{\sin\phi\cos\phi}{r^2} \frac{\PD}{\PD \phi}
-\frac{2\sin\phi\cos\phi}{r} \frac{\PD}{\PD r}\frac{\PD}{\PD \phi}\\
&\ \ \ 
+\frac{\cos^2\theta\cos^2\phi}{r} \frac{\PD}{\PD r}
+\frac{-\sin\theta\cos\theta\cos^2\phi}{r^2}\frac{\PD}{\PD \theta}
+\frac{\cos^2\theta\cos^2\phi}{r^2} \frac{\PD^2}{\PD \theta^2}
+\frac{\cos^2\theta\sin\phi\cos\phi}{r^2\sin^2\theta} \frac{\PD}{\PD \phi}\\
&\ \ \ 
-\frac{2\cos\theta\sin\phi\cos\phi}{r^2\sin\theta} \frac{\PD}{\PD \theta}\frac{\PD}{\PD \phi}
+\frac{\sin^2\phi}{r} \frac{\PD}{\PD r}
+\frac{\cos\theta\sin^2\phi}{r^2\sin\theta} \frac{\PD}{\PD \theta}
+\frac{\sin\phi\cos\phi}{r^2\sin^2\theta} \frac{\PD}{\PD \phi}
+\frac{\sin^2\phi}{r^2\sin^2\theta} \frac{\PD^2}{\PD \phi^2}\\

&\ \ \ 
+\sin^2\theta\sin^2\phi \frac{\PD^2}{\PD r^2}
-\frac{\sin\theta\cos\theta\sin^2\phi}{r^2} \frac{\PD}{\PD \theta}
+\frac{2\sin\theta\cos\theta\sin^2\phi}{r} \frac{\PD}{\PD r}\frac{\PD}{\PD \theta}
-\frac{\sin\phi\cos\phi}{r^2} \frac{\PD}{\PD \phi}
+\frac{2\sin\phi\cos\phi}{r} \frac{\PD}{\PD r}\frac{\PD}{\PD \phi}\\
&\ \ \ 
+\frac{\cos^2\theta\sin^2\phi}{r} \frac{\PD}{\PD r}
+\frac{-\sin\theta\cos\theta\sin^2\phi}{r^2} \frac{\PD}{\PD \theta}
+\frac{\cos^2\theta\sin^2\phi}{r^2} \frac{\PD^2}{\PD \theta^2}
-\frac{\cos^2\theta\sin\phi\cos\phi}{r^2\sin^2\theta} \frac{\PD}{\PD \phi}
+\frac{2\cos\theta\sin\phi\cos\phi}{r^2\sin\theta} \frac{\PD}{\PD \theta}\frac{\PD}{\PD \phi}\\
&\ \ \ 
+\frac{\cos^2\phi}{r} \frac{\PD}{\PD r}
+\frac{\cos\theta\cos^2\phi}{r^2\sin\theta} \frac{\PD}{\PD \theta}
+\frac{-\sin\phi\cos\phi}{r^2\sin^2\theta} \frac{\PD}{\PD \phi}
+\frac{\cos^2\phi}{r^2\sin^2\theta} \frac{\PD^2}{\PD \phi^2}\\

&\ \ \ 
+\cos^2\theta \frac{\PD^2}{\PD r^2}
+\frac{\sin\theta\cos\theta}{r^2} \frac{\PD}{\PD \theta}
-\frac{2\sin\theta\cos\theta}{r} \frac{\PD}{\PD r}\frac{\PD}{\PD \theta}
\\
&\ \ \ 
-\frac{\sin^2\theta}{r} \frac{\PD}{\PD r}-
-\frac{\sin\theta\cos\theta}{r^2} \frac{\PD}{\PD \theta}
-\frac{\sin^2\theta}{r^2} \frac{\PD^2}{\PD \theta^2}\\
);
&math(
&=
(\sin^2\theta\cos^2\phi+\sin^2\theta\sin^2\phi+\cos^2\theta)\frac{\PD^2}{\PD r^2}
\\&\ \ \ 
+\Big(\frac{\cos^2\theta\cos^2\phi}{r}+\frac{\sin^2\phi}{r}+\frac{\cos^2\theta\sin^2\phi}{r}+\frac{\cos^2\phi}{r} +\frac{\sin^2\theta}{r}\Big) \frac{\PD}{\PD r}
\\&\ \ \ 
+\Big(-\cancel{\frac{\sin\theta\cos\theta\cos^2\phi}{r^2}}+\cancel{\frac{-\sin\theta\cos\theta\cos^2\phi}{r^2}}+\frac{\cos\theta\sin^2\phi}{r^2\sin\theta}-\cancel{\frac{\sin\theta\cos\theta\sin^2\phi}{r^2}}\\
&\hspace{4cm}+\cancel{\frac{-\sin\theta\cos\theta\sin^2\phi}{r^2}}+\frac{\cos\theta\cos^2\phi}{r^2\sin\theta}
+\cancel{\frac{\sin\theta\cos\theta}{r^2}}+\cancel{\frac{\sin\theta\cos\theta}{r^2}}\Big) \frac{\PD}{\PD \theta}
\\&\ \ \ 
+\Big(\frac{\cos^2\theta\cos^2\phi}{r^2}+\frac{\cos^2\theta\sin^2\phi}{r^2}+\frac{\sin^2\theta}{r^2}\Big) \frac{\PD^2}{\PD \theta^2}
\\&\ \ \ 
+\Big(\cancel{\frac{\sin\phi\cos\phi}{r^2}}+\cancel{\frac{\cos^2\theta\sin\phi\cos\phi}{r^2\sin^2\theta}}+\cancel{\frac{\sin\phi\cos\phi}{r^2\sin^2\theta}}-\cancel{\frac{\sin\phi\cos\phi}{r^2}}-\cancel{\frac{\cos^2\theta\sin\phi\cos\phi}{r^2\sin^2\theta}}-\cancel{\frac{\sin\phi\cos\phi}{r^2\sin^2\theta}}\Big) \frac{\PD}{\PD \phi}
\\&\ \ \ 
+\Big(\frac{\sin^2\phi}{r^2\sin^2\theta}+\frac{\cos^2\phi}{r^2\sin^2\theta}\Big) \frac{\PD^2}{\PD \phi^2}
\\&\ \ \ 
+\Big(\cancel{\frac{2\sin\theta\cos\theta\cos^2\phi}{r}}+\cancel{\frac{2\sin\theta\cos\theta\sin^2\phi}{r}}-\cancel{\frac{2\sin\theta\cos\theta}{r}}\Big) \frac{\PD}{\PD r}\frac{\PD}{\PD \theta}
\\&\ \ \ 
+\Big(-\cancel{\frac{2\sin\phi\cos\phi}{r}}+\cancel{\frac{2\sin\phi\cos\phi}{r}}\Big) \frac{\PD}{\PD r}\frac{\PD}{\PD \phi}
\\&\ \ \ 
+\Big(-\cancel{\frac{2\cos\theta\sin\phi\cos\phi}{r^2\sin\theta}}+\cancel{\frac{2\cos\theta\sin\phi\cos\phi}{r^2\sin\theta}}\Big)\frac{\PD}{\PD \theta}\frac{\PD}{\PD \phi}\\

&=
\frac{\PD^2}{\PD r^2}+\frac{2}{r} \frac{\PD}{\PD r}+\frac{\cos\theta}{r^2\sin\theta} \frac{\PD}{\PD \theta}
+\frac{1}{r^2}\frac{\PD^2}{\PD \theta^2}+\frac{1}{r^2\sin^2\theta} \frac{\PD^2}{\PD \phi^2}
\\
&=
\frac{\PD^2}{\PD r^2}+\frac{2}{r} \frac{\PD}{\PD r}
+\frac{1}{r^2}\underbrace{\bigg[\frac{1}{\sin\theta} \frac{\PD}{\PD \theta}
\Big(\sin\theta\frac{\PD}{\PD \theta}\Big)+\frac{1}{\sin^2\theta} \frac{\PD^2}{\PD \phi^2}\bigg]}_{=\,\Lambda}
);

Counter: 959 (from 2010/06/03), today: 1, yesterday: 0