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シュレーディンガー方程式を解く4〜ラプラシアン2〜

前口上です(・∀・)

ラプラシアン極座標変換ということであれば,それを示している媒体は天下にゴマンとございますじゃ.

このブログはお勉強ノートでして,ゆえに,わちきらは作業過程を示さねばならん.
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{
\begin{align}
\left(\displaystyle\frac{\partial}{\partial x}\cdot\displaystyle\frac{\partial}{\partial x}\right)&=\left(\sin{\theta}\cos{\phi}\frac{\partial}{\partial r}+\frac{1}{r}\cos{\theta}\cos{\phi}\frac{\partial}{\partial\theta}-\frac{\sin{\phi}}{r\sin{\theta}}\frac{\partial}{\partial\phi}\right)\left(\sin{\theta}\cos{\phi}\frac{\partial}{\partial r}+\frac{1}{r}\cos{\theta}\cos{\phi}\frac{\partial}{\partial\theta}-\frac{\sin{\phi}}{r\sin{\theta}}\frac{\partial}{\partial\phi}\right)\\
&=\sin{\theta}\cos{\phi}\frac{\partial}{\partial r}\sin{\theta}\cos{\phi}\frac{\partial}{\partial r}+\sin{\theta}\cos{\phi}\frac{\partial}{\partial r}\frac{1}{r}\cos{\theta}\cos{\phi}\frac{\partial}{\partial\theta}+\sin{\theta}\cos{\phi}\frac{\partial}{\partial r}\left(-\frac{\sin{\phi}}{r\sin{\theta}}\frac{\partial}{\partial\phi}\right)\\
&\ \ \ \ +\frac{1}{r}\cos{\theta}\cos{\phi}\frac{\partial}{\partial\theta}\sin{\theta}\cos{\phi}\frac{\partial}{\partial r}+\frac{1}{r}\cos{\theta}\cos{\phi}\frac{\partial}{\partial\theta}\frac{1}{r}\cos{\theta}\cos{\phi}\frac{\partial}{\partial\theta}+\frac{1}{r}\cos{\theta}\cos{\phi}\frac{\partial}{\partial\theta}\left(-\frac{\sin{\phi}}{r\sin{\theta}}\frac{\partial}{\partial\phi}\right)\\
&\ \ \ \ +\left(-\frac{\sin{\phi}}{r\sin{\theta}}\frac{\partial}{\partial\phi}\right)\sin{\theta}\cos{\phi}\frac{\partial}{\partial r}+\left(-\frac{\sin{\phi}}{r\sin{\theta}}\frac{\partial}{\partial\phi}\right)\frac{1}{r}\cos{\theta}\cos{\phi}\frac{\partial}{\partial\theta}+\left(-\frac{\sin{\phi}}{r\sin{\theta}}\frac{\partial}{\partial\phi}\right)\left(-\frac{\sin{\phi}}{r\sin{\theta}}\frac{\partial}{\partial\phi}\right)\\
\\
&=\sin^2{\theta}\cos^2{\phi}\frac{\partial^2}{\partial r^2}+\sin{\theta}\cos{\theta}\cos^2{\phi}\left(\frac{\partial}{\partial r}\frac{1}{r}\frac{\partial}{\partial\theta}+\frac{1}{r}\frac{\partial}{\partial r}\frac{\partial}{\partial\theta}\right)-\sin{\phi}\cos{\phi}\left(\frac{\partial}{\partial r}\frac{1}{r}\frac{\partial}{\partial\phi}+\frac{1}{r}\frac{\partial}{\partial r}\frac{\partial}{\partial\phi}\right)\\&\ \ \ \ +\frac{1}{r}\cos{\theta}\cos^2{\phi}\left(\frac{\partial}{\partial\theta}\sin{\theta}\frac{\partial}{\partial r}+\sin{\theta}\frac{\partial}{\partial\theta}\frac{\partial}{\partial r}\right)+\frac{1}{r^2}\cos{\theta}\cos^2{\phi}\left(\frac{\partial}{\partial\theta}\cos{\theta}\frac{\partial}{\partial\theta}+\cos{\theta}\frac{\partial}{\partial\theta}\frac{\partial}{\partial\theta}\right)-\frac{1}{r^2}\cos{\theta}\sin{\phi}\cos{\phi}\left(\frac{\partial}{\partial\theta}\frac{1}{\sin{\theta}}\frac{\partial}{\partial\phi}+\frac{1}{\sin{\theta}}\frac{\partial}{\partial\theta}\frac{\partial}{\partial\phi}\right)\\
&\ \ \ \ -\frac{1}{r}\sin{\phi}\left(\frac{\partial}{\partial\phi}\cos{\phi}\frac{\partial}{\partial r}+\cos{\phi}\frac{\partial}{\partial\phi}\frac{\partial}{\partial r}\right)-\frac{\sin{\phi}\cos{\theta}}{r^2\sin{\theta}}\left(\frac{\partial}{\partial\phi}\cos{\phi}\frac{\partial}{\partial\theta}+\cos{\phi}\frac{\partial}{\partial\phi}\frac{\partial}{\partial\theta}\right)+\frac{\sin{\phi}}{r^2\sin^2{\theta}}\left(\frac{\partial}{\partial\phi}\sin{\phi}\frac{\partial}{\partial\phi}+\sin{\phi}\frac{\partial}{\partial\phi}\frac{\partial}{\partial\phi}\right)\\
\\
&=\sin^2{\theta}\cos^2{\phi}\frac{\partial^2}{\partial r^2}-\frac{1}{r^2}\sin{\theta}\cos{\theta}\cos^2{\phi}\frac{\partial}{\partial\theta}+\frac{1}{r}\sin{\theta}\cos{\theta}\cos^2{\phi}\frac{\partial^2}{\partial r\partial\theta}+\frac{1}{r^2}\sin{\phi}\cos{\phi}\frac{\partial}{\partial\phi}-\frac{1}{r}\sin{\phi}\cos{\phi}\frac{\partial^2}{\partial r\partial\theta}\\
&\ \ \ \ +\frac{1}{r}\cos^2{\theta}\cos^2{\phi}\frac{\partial}{\partial r}+\frac{1}{r}\sin{\theta}\cos{\theta}\cos^2{\phi}\frac{\partial^2}{\partial r\partial\theta}-\frac{1}{r^2}\sin{\theta}\cos{\theta}\cos^2{\phi}\frac{\partial}{\partial\theta}+\frac{1}{r^2}\cos^2{\theta}\cos^2{\phi}\frac{\partial^2}{\partial\theta^2}+\frac{1}{r^2}\frac{\cos^2{\theta}\sin{\phi}\cos{\phi}}{\sin^2{\theta}}\frac{\partial}{\partial\phi}\\
&\ \ \ \ -\frac{1}{r^2}\frac{\cos{\theta}\sin{\phi}\cos^2{\phi}}{\sin{\theta}}\frac{\partial^2}{\partial\theta\partial\phi}+\frac{\sin^2{\phi}}{r}\frac{\partial}{\partial r}-\frac{1}{r}\sin{\phi}\cos{\phi}\frac{\partial^2}{\partial r\partial\phi}+\frac{1}{r^2}\frac{\sin^2{\phi}\cos{\theta}}{\sin{\theta}}\frac{\partial}{\partial\theta}-\frac{1}{r^2}\frac{\sin{\phi}\cos{\phi}\cos{\theta}}{\sin{\theta}}\frac{\partial^2}{\partial\theta\partial\phi}\\
&\ \ \ \ +\frac{1}{r^2}\frac{\sin{\phi}\cos{\phi}}{\sin^2{\theta}}\frac{\partial}{\partial\phi}+\frac{1}{r^2}\frac{\sin^2{\phi}}{\sin^2{\theta}}\frac{\partial^2}{\partial\phi^2}\\
\\
&=\sin^2{\theta}\cos^2{\phi}\frac{\partial^2}{\partial r^2}-\frac{2}{r^2}\sin{\theta}\cos{\theta}\cos^2{\phi}\frac{\partial}{\partial\theta}+\frac{2}{r}\sin{\theta}\cos{\theta}\cos^2{\phi}\frac{\partial^2}{\partial r\partial\theta}+\frac{1}{r^2}\sin{\phi}\cos{\phi}\frac{\partial}{\partial\phi}\\
&\ \ \ \ -\frac{2}{r}\sin{\phi}\cos{\phi}\frac{\partial^2}{\partial r\partial\theta}+\frac{1}{r}\cos^2{\theta}\cos^2{\phi}\frac{\partial}{\partial r}+\frac{1}{r^2}\cos^2{\theta}\cos^2{\phi}\frac{\partial^2}{\partial\theta^2}+\frac{1}{r^2}\frac{\cos^2{\theta}\sin{\phi}\cos{\phi}}{\sin^2{\theta}}\frac{\partial}{\partial\phi}-\frac{2}{r^2}\frac{\cos{\theta}\sin{\phi}\cos{\phi}}{\sin{\theta}}\frac{\partial^2}{\partial\theta\partial\phi}\\
&\ \ \ \ +\frac{\sin^2{\phi}}{r}\frac{\partial}{\partial r}+\frac{1}{r^2}\frac{\sin^2{\phi}\cos{\theta}}{\sin{\theta}}\frac{\partial}{\partial\theta}+\frac{1}{r^2}\frac{\sin{\phi}\cos{\phi}}{\sin^2{\theta}}\frac{\partial}{\partial\phi}+\frac{1}{r^2}\frac{\sin^2{\phi}}{\sin^2{\theta}}\frac{\partial^2}{\partial\phi^2}
\end{align}
}


(;ω;)
ほロほろ…暑さもあってちょっと気分悪くなりましただ.

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