ラプラシアンΔの極座標表示を導く計算方法

シュレディンガー方程式のハミルトニアンに含まれるラプラシアンΔを極座標に変換するときの計算をおこなう。ラプラシアンに２階微分が入っているため、 $(x,y,z)\rightarrow(r,\theta,\phi)$ の変換は手間がかかることを知ってもらいたい。よく見る Δ の極座標表示は以下の通り。

参考：球面調和関数、ルジャンドル多項式、ラーゲル多項式のざっくりまとめ

ラプラシアンΔの極座標表示

$\begin{eqnarray*} \Delta&=&\frac{\partial^2}{\partial r^2} +\frac{2}{r}\,\frac{\partial}{\partial r}+\frac{1}{r^2}\,\Lambda(\theta,\phi)\\ \\ \Lambda(\theta,\phi)&=& \frac{\partial^2}{\partial\theta^2}+\frac{\cos{\theta}}{\sin{\theta}}\,\frac{\partial}{\partial \theta}+\frac{1}{\sin^2{\theta}}\,\frac{\partial^2}{\partial \phi^2} \end{eqnarray*}$

1. １階微分の極座標表示
2. ラプラシアンΔの極座標表示
3. まとめ

1. １階微分の極座標表示

　１階微分の極座標表示の導出過程は以下を参考にすると良い。

∂/∂x、∂/∂y、∂/∂z を極座標表示に変換する計算

　ここでは１階微分については最終的な結果だけ示しておく。

１階微分の極座標表示

$\begin{eqnarray*} \frac{\partial}{\partial x}&=&\sin{\theta}\cos{\phi} \frac{\partial}{\partial r} +\frac{1}{r}\cos{\theta}\cos{\phi} \frac{\partial}{\partial \theta} -\frac{1}{r}\frac{\sin{\phi}}{\sin{\theta}} \frac{\partial}{\partial \phi}\\\\ \frac{\partial}{\partial y}&=& \sin{\theta}\sin{\phi} \frac{\partial}{\partial r} +\frac{1}{r}\cos{\theta}\sin{\phi} \frac{\partial}{\partial \theta} +\frac{1}{r}\frac{\cos{\phi}}{\sin{\theta}} \frac{\partial}{\partial \phi}\\ \\ \frac{\partial}{\partial z}&=& \cos{\theta} \frac{\partial}{\partial r} -\frac{1}{r}\sin{\theta} \frac{\partial}{\partial \theta} \end{eqnarray*}$

2. ラプラシアンΔの極座標表示

　ラプラシアンの $x,y,z$ 表示は、

$\begin{eqnarray*} \Delta = \frac{\partial^2}{\partial x^2} +\frac{\partial^2}{\partial y^2} +\frac{\partial^2}{\partial z^2} \end{eqnarray*}$

であり、２階偏微分を含むため変換は厄介である。導出は、

１階偏微分の変換（項目1で示した）
２階偏微分の変換

の手順で計算する。

2.1 ∂^2/∂x^2

$\frac{\partial}{\partial x}$ を2回使う。

$\begin{eqnarray*} \frac{\partial^2}{\partial x^2} &=&\sin{\theta}\cos{\phi}\frac{\partial}{\partial r}\left( \sin{\theta}\cos{\phi} \frac{\partial}{\partial r} +\frac{1}{r}\cos{\theta}\cos{\phi} \frac{\partial}{\partial \theta} -\frac{1}{r}\frac{\sin{\phi}}{\sin{\theta}} \frac{\partial}{\partial \phi}\right)\\ \\ &+& \frac{1}{r}\cos{\theta}\cos{\phi} \frac{\partial}{\partial \theta} \left( \sin{\theta}\cos{\phi} \frac{\partial}{\partial r} +\frac{1}{r}\cos{\theta}\cos{\phi} \frac{\partial}{\partial \theta} -\frac{1}{r}\frac{\sin{\phi}}{\sin{\theta}} \frac{\partial}{\partial \phi}\right)\\ \\ &-& \frac{1}{r}\frac{\sin{\phi}}{\sin{\theta}} \frac{\partial}{\partial \phi} \left( \sin{\theta}\cos{\phi} \frac{\partial}{\partial r} +\frac{1}{r}\cos{\theta}\cos{\phi} \frac{\partial}{\partial \theta} -\frac{1}{r}\frac{\sin{\phi}}{\sin{\theta}} \frac{\partial}{\partial \phi}\right) \end{eqnarray*}$

1行目

$\begin{eqnarray*} &&\sin{\theta}\cos{\phi}\frac{\partial}{\partial r}\left( \sin{\theta}\cos{\phi} \frac{\partial}{\partial r} +\frac{1}{r}\cos{\theta}\cos{\phi} \frac{\partial}{\partial \theta} -\frac{1}{r}\frac{\sin{\phi}}{\sin{\theta}} \frac{\partial}{\partial \phi}\right)\\ \\&=& \sin^2{\theta}\cos^2{\phi}\frac{\partial^2}{\partial r^2} \\ \\ &-& \frac{1}{r^2}\cos{\theta}\cos{\phi} \frac{\partial}{\partial \theta} + \frac{1}{r}\cos{\theta}\cos{\phi} \frac{\partial^2}{\partial r \partial\theta}\\ \\ &+& \frac{1}{r^2}\frac{\sin{\phi}}{\sin{\theta}} \frac{\partial}{\partial \phi} -\frac{1}{r}\frac{\sin{\phi}}{\sin{\theta}} \frac{\partial^2}{\partial r\partial \phi} \end{eqnarray*}$

間違えやすいポイント

間違えがちなポイント

$\begin{eqnarray*} \frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial}{\partial \theta}\right)=-\frac{1}{r^2}\frac{\partial}{\partial \theta}+\frac{1}{r}\frac{\partial^2}{\partial r \partial \theta} \end{eqnarray*}$

これは適当な関数 $\Phi(r,\theta,\phi)$ を用意すればわかると思う。

$\begin{eqnarray*} \frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial \Phi(r,\theta,\phi)}{\partial \theta}\right)=-\frac{1}{r^2}\frac{\partial \Phi(r,\theta,\phi)}{\partial \theta}+\frac{1}{r}\frac{\partial^2 \Phi(r,\theta,\phi)}{\partial r \partial \theta} \end{eqnarray*}$

何が $r$ の関数になっているところがはっきりしていれば間違えにくい。

2行目

$\begin{eqnarray*} &&\frac{1}{r}\cos{\theta}\cos{\phi} \frac{\partial}{\partial \theta} \left( \sin{\theta}\cos{\phi} \frac{\partial}{\partial r} +\frac{1}{r}\cos{\theta}\cos{\phi} \frac{\partial}{\partial \theta} -\frac{1}{r}\frac{\sin{\phi}}{\sin{\theta}} \frac{\partial}{\partial \phi}\right)\\ &=& \frac{1}{r}\cos^2{\theta}\cos^2{\phi} \frac{\partial}{\partial r} + \frac{1}{r}\sin{\theta}\cos{\theta}\cos^2{\phi} \frac{\partial^2}{\partial r \partial \theta}\\ \\ &-& \frac{1}{r^2}\sin{\theta}\cos{\theta}\cos^2{\phi} \frac{\partial}{\partial \theta} +\frac{1}{r^2}\cos^2{\theta}\cos^2{\phi} \frac{\partial^2}{\partial \theta^2} \\ \\ &+& \frac{1}{r^2}\frac{ \cos^2{\theta }\cos{\phi} \sin{\phi} }{ \sin^2{\theta} } \frac{\partial}{\partial \phi} -\frac{1}{r^2} \frac{\cos{\theta}\cos{\phi}\sin{\phi}}{\sin^2{\theta}} \frac{\partial^2}{\partial \phi^2} \end{eqnarray*}$

3行目

$\begin{eqnarray*} &-&\frac{1}{r}\frac{\sin{\phi}}{\sin{\theta}} \frac{\partial}{\partial \phi} \left( \sin{\theta}\cos{\phi} \frac{\partial}{\partial r} +\frac{1}{r}\cos{\theta}\cos{\phi} \frac{\partial}{\partial \theta} -\frac{1}{r}\frac{\sin{\phi}}{\sin{\theta}} \frac{\partial}{\partial \phi}\right)\\ \\ &=& \frac{1}{r}\sin^2{\phi} \frac{\partial}{\partial r} -\frac{1}{r}\sin{\phi}\cos{\phi} \frac{\partial^2}{\partial r \partial \phi}\\ \\ &+& \frac{1}{r^2}\frac{\cos{\theta}\sin^2{\phi}}{\sin{\theta}} \frac{\partial}{\partial \theta} - \frac{1}{r^2}\frac{\cos{\theta}\sin{\phi}\cos{\phi}}{\sin{\theta}} \frac{\partial^2}{\partial \theta \partial \phi} \\ \\ &+& \frac{1}{r^2}\frac{\sin{\phi}\cos{\phi}}{\sin^2{\theta}} \frac{\partial}{\partial \phi} + \frac{1}{r^2}\frac{\sin^2{\phi}}{\sin^2{\theta}} \frac{\partial^2}{\partial \phi^2} \end{eqnarray*}$

2.2 ∂^2/∂y^2

　同様に計算する。

$\begin{eqnarray*} \frac{\partial^2}{\partial x^2}&=& \sin{\theta}\sin{\phi} \frac{\partial}{\partial r} \left( \sin{\theta}\sin{\phi} \frac{\partial}{\partial r} +\frac{1}{r}\cos{\theta}\sin{\phi} \frac{\partial}{\partial \theta} +\frac{1}{r}\frac{\cos{\phi}}{\sin{\theta}} \frac{\partial}{\partial \phi} \right) \\ \\ &+&\frac{1}{r}\cos{\theta}\sin{\phi} \frac{\partial}{\partial \theta}\left( +\sin{\theta}\sin{\phi} \frac{\partial}{\partial r} +\frac{1}{r}\cos{\theta}\sin{\phi} \frac{\partial}{\partial \theta} +\frac{1}{r}\frac{\cos{\phi}}{\sin{\theta}} \frac{\partial}{\partial \phi} \right) \\ \\ &+&\frac{1}{r}\cos{\phi}\sin{\theta} \frac{\partial}{\partial \phi}\left( \sin{\theta}\sin{\phi} \frac{\partial}{\partial r} +\frac{1}{r}\cos{\theta}\sin{\phi} \frac{\partial}{\partial \theta} +\frac{1}{r}\frac{\cos{\phi}}{\sin{\theta}} \frac{\partial}{\partial \phi} \right) \end{eqnarray*}$

1行目

$\begin{eqnarray*} && \sin{\phi}\sin{\theta} \frac{\partial}{\partial r} \left( \sin{\phi}\sin{\theta} \frac{\partial}{\partial r} +\frac{1}{r}\cos{\theta}\sin{\phi} \frac{\partial}{\partial \theta} +\frac{1}{r}\frac{\cos{\phi}}{\sin{\theta}} \frac{\partial}{\partial \phi} \right) \\ \\ &=& \sin^2{\theta}\sin^2{\phi} \frac{\partial^2}{\partial r^2} -\frac{1}{r^2}\sin{\theta}\cos{\theta}\sin^2{\phi} \frac{\partial}{\partial \theta}\\ \\ &+& \frac{1}{r}\sin{\theta}\cos{\theta}\sin^2{\phi} \frac{\partial}{\partial \theta} + \frac{1}{r}\sin{theta}\cos{\theta}\sin^2{\phi} \frac{\partial^2}{\partial r \partial\theta}\\ \\ &-& \frac{1}{r^2}\sin{\phi}\cos{\phi} \frac{\partial}{\partial \phi} + \frac{1}{r}\sin{\phi}\cos{\phi} \frac{\partial^2}{\partial r \partial \phi} \end{eqnarray*}$

2行目

$\begin{eqnarray*} && \frac{1}{r}\cos{\theta}\sin{\phi} \frac{\partial}{\partial \theta}\left( +\sin{\theta}\sin{\phi} \frac{\partial}{\partial r} +\frac{1}{r}\cos{\theta}\sin{\phi} \frac{\partial}{\partial \theta} +\frac{1}{r}\frac{\cos{\phi}}{\sin{\theta}} \frac{\partial}{\partial \phi} \right) \\ \\ &=& \frac{1}{r} \cos^2{\theta}\sin^2{\phi} \frac{\partial}{\partial r} +\frac{1}{r}\sin{\theta}\cos{\theta}\sin^2{\phi} \frac{\partial^2}{\partial r \partial \theta} \\ \\ &-& \frac{1}{r^2}\sin{\theta}\cos{\theta}\sin^2{\phi} \frac{\partial}{\partial \theta} + \frac{1}{r^2}\cos^2{\theta}\sin^2{\phi} \frac{\partial^2}{\partial \theta^2} \\ \\ &-& \frac{1}{r^2}\frac{\cos^2{\theta}\cos{\phi}\sin{\phi}}{\sin^2{\theta}} \frac{\partial}{\partial \phi} + \frac{1}{r^2}\frac{\cos{\theta}\sin{\phi}\cos{\phi}}{\sin{\theta}} \frac{\partial^2}{\partial \theta \partial \phi} \end{eqnarray*}$

3行目

$\begin{eqnarray*} && \frac{1}{r}\cos{\phi}\sin{\theta} \frac{\partial}{\partial \phi}\left( \sin{\theta}\sin{\phi} \frac{\partial}{\partial r} +\frac{1}{r}\cos{\theta}\sin{\phi} \frac{\partial}{\partial \theta} +\frac{1}{r}\frac{\cos{\phi}}{\sin{\theta}} \frac{\partial}{\partial \phi} \right) \\ \\ &=& \frac{1}{r}\cos^2{\phi} \frac{\partial}{\partial r} + \frac{1}{r}\sin{\phi}\cos{\phi} \frac{\partial^2}{\partial r \partial \phi}\\ \\ &+& \frac{1}{r^2}\frac{\cos{\theta}\cos^2{\phi}}{\sin{\theta}} \frac{\partial}{\partial \theta} +\frac{1}{r^2}\frac{\cos{\theta}\sin{\phi}\cos{\phi}}{\sin{\theta}} \frac{\partial^2}{\partial \theta \partial \phi} \\ \\ &-& \frac{1}{r^2}\frac{\sin{\phi}\cos{\phi}}{\sin^2{\theta}} \frac{\partial}{\partial \phi} + \frac{1}{r^2}\frac{\cos^2{\phi}}{\sin^2{\theta}} \frac{\partial^2}{\partial \phi^2} \end{eqnarray*}$

2.3 ∂^2/∂z^2

$\begin{eqnarray*} \frac{\partial^2}{\partial z^2} &=& \cos{\theta} \frac{\partial}{\partial r}\left( \cos{\theta} \frac{\partial}{\partial r} -\frac{1}{r}\sin{\theta} \frac{\partial}{\partial \theta} \right)\\ \\ &-&\frac{1}{r}\sin{\theta} \frac{\partial}{\partial \theta} \left( \cos{\theta} \frac{\partial}{\partial r} -\frac{1}{r}\sin{\theta} \frac{\partial}{\partial \theta} \right) \end{eqnarray*}$

1行目

$\begin{eqnarray*} &&\cos{\theta} \frac{\partial}{\partial r}\left( \cos{\theta} \frac{\partial}{\partial r} -\frac{1}{r}\sin{\theta} \frac{\partial}{\partial \theta} \right) \\ \\ &=& \cos^2{\theta}\frac{\partial^2}{\partial r^2} \\ \\ &+& \frac{1}{r^2}\cos{\theta}\sin{\theta} \frac{\partial}{\partial \theta} -\frac{1}{r}\cos{\theta}\sin{\theta} \frac{\partial^2}{\partial r \partial \theta} \end{eqnarray*}$

2行目

$\begin{eqnarray*} &&\frac{1}{r}\sin{\theta} \frac{\partial}{\partial \theta} \left( \cos{\theta} \frac{\partial}{\partial r} -\frac{1}{r}\sin{\theta} \frac{\partial}{\partial \theta} \right)\\ \\ &=& \frac{1}{r}\sin{\theta}\frac{\partial}{\partial r} -\frac{1}{r}\sin\theta \cos\theta \\ \\ &+& \frac{1}{r^2}\sin\theta\cos\theta \frac{\partial}{\partial \theta} +\frac{1}{r^2}\sin^2{\theta} \frac{\partial^2}{\partial \theta^2} \end{eqnarray*}$

2.4 ∂^2/∂x^2 + ∂^2/∂y^2 + ∂^2/∂z^2

方針：項ごとに係数を計算して、まとめる。

$\frac{\partial^2}{\partial r^2}$ ：

$\begin{eqnarray*} \sin^2{\theta}\cos^2{\phi}+\sin^2{\theta}\sin^2{\phi} +\cos^2{\theta}= \textcolor{red}{1} \end{eqnarray*}$

$\frac{\partial}{\partial r}$ ：

$\begin{eqnarray*} &+&\frac{1}{r}\cos^2{\theta}\cos^2{\phi} +\frac{1}{r}\sin^2{\phi}+\frac{1}{r} \cos^2{\theta}\sin^2{\phi}+ \frac{1}{r}\cos^2{\phi} +\frac{1}{r}\sin^2{\theta} \\ \\ &=& \frac{1}{r}\cos^2{\theta}+\frac{1}{r}+\frac{1}{r}\sin^2{\theta} \\ \\ &=& \textcolor{red}{\frac{2}{r}} \end{eqnarray*}$

$\frac{\partial^2}{\partial r \partial \theta}$ ：

$\begin{eqnarray*} &+&\frac{1}{r}\sin\theta \cos\theta \cos^2{\phi} +\frac{1}{r}\sin\theta \cos\theta \cos^2{\phi} + \frac{1}{r}\sin\theta \cos\theta \sin^2{\phi}\\ &+&\frac{1}{r}\sin\theta \cos\theta \sin^2{\phi} -\frac{1}{r}\sin\theta \cos\theta - \frac{1}{r}\sin\theta \cos\theta\\ \\ &=&\textcolor{red}{0} \end{eqnarray*}$

$\frac{\partial^2}{\partial r \partial \phi}$ ：

$\begin{eqnarray*} &-&\frac{1}{r}\sin\phi \cos\phi - \frac{1}{r}\sin\phi \cos\phi +\frac{1}{r}\sin\phi \cos\phi + \frac{1}{r} \cos\phi \sin\phi \\ \\ &=&\textcolor{red}{0} \end{eqnarray*}$

$\frac{\partial^2}{\partial \theta^2}$ ：

$\begin{eqnarray*} &+&\frac{1}{r^2}\cos^2{\theta}\cos^2{\phi} +\frac{1}{r^2}\cos^2\theta \sin^2{\phi}+\frac{1}{r^2}\sin^2{\theta} \\ \\ &=&\textcolor{red}{\frac{1}{r^2}} \end{eqnarray*}$

$\frac{\partial}{\partial \theta}$ ：

$\begin{eqnarray*} &-&\frac{1}{r^2}\sin\theta \cos\theta \cos^2{\phi} -\frac{1}{r^2}\cos\theta \sin\theta \cos^2{\phi} +\frac{1}{r^2}\frac{\cos\theta \sin^2{\phi}}{\sin\theta}\\ &-&\frac{1}{r^2}\sin\theta \cos\theta \sin^2{\phi} -\frac{1}{r^2}\cos\theta \sin\theta \sin^2{\phi} +\frac{1}{r^2}\frac{\cos\theta \cos^2{\phi}}{\sin\theta}\\ &+&\frac{1}{r^2}\cos\theta +\frac{1}{r^2}\sin\theta \cos\theta \\ \\ &=& -\frac{2}{r^2}\sin\theta \cos\theta + \frac{1}{r^2}\frac{\cop{\theta}}{\sin{\theta}}+\frac{2}{r^2}\cos\theta\sin\theta \\ \\ &=&\textcolor{red}{\frac{1}{r^2}\frac{\cos\theta}{\sin\theta}} \end{eqnarray*}$

$\frac{\partial^2}{\partial \theta \partial \phi}$ ：

$\begin{eqnarray*} &-&\frac{1}{r^2}\frac{\cos\theta \cos\phi \sin\phi}{\sin\theta} -\frac{1}{r^2}\frac{\cos\theta \sin\phi \cos\phi}{\sin\theta} \\ &+&\frac{1}{r^2}\frac{\cos\theta \sin\phi \cos\phi}{\sin\theta} \frac{1}{r^2}\frac{\cos\theta \sin\phi \cos\phi}{\sin\theta} \\ \\ &=&\textcolor{red}{0} \end{eqnarray*}$

$\frac{\partial^2}{\partial \phi^2}$ ：

$\begin{eqnarray*} &+&\frac{1}{r^2}\frac{\sin^2{\phi}}{\sin^2{\theta}}+\frac{1}{r^2}\frac{\cos^2{\phi}}{\sin^2{\theta}}\\ \\ &=&\textcolor{red}{\frac{1}{r^2}\frac{1}{\sin^2{\theta}}} \end{eqnarray*}$

$\frac{\partial}{\partial \phi}$ ：

$\begin{eqnarray*} &+&\frac{1}{r^2}\sin\phi\cos\phi +\frac{1}{r^2}\frac{\cos^2{\theta} \cos\phi \sin\phi}{\sin^2{\theta}}+\frac{1}{r^2}\frac{\sin\phi \cos\phi}{\sin^2{\theta}}\\ &-&\frac{1}{r^2}\sin\phi \cos\phi -\frac{1}{r^2}\frac{\cos^2{\theta}\cos\phi \sin\phi}{\sin^2{\theta}} -\frac{1}{r^2}\frac{\sin\phi \cos\phi}{\sin^2{\theta}} \\ \\ &=& \textcolor{red}{0} \end{eqnarray*}$

2.5 全部足す

　以上をまとめることで、ラプラシアンの極座標表示が求められる。

ラプラシアンΔの極座標表示

3. まとめ

　大変な計算量でした。お疲れ様でした。途中の「間違えやすいポイント」にさえ注意すれば、ラプラシアンの極座標表示は求められます。

【微分】ラプラシアンΔの極座標表示を導く計算

1. １階微分の極座標表示

2. ラプラシアンΔの極座標表示

2.1 ∂^2/∂x^2

1行目

間違えやすいポイント

2行目

3行目

2.2 ∂^2/∂y^2

1行目

2行目

3行目

2.3 ∂^2/∂z^2

1行目

2行目

2.4 ∂^2/∂x^2 + ∂^2/∂y^2 + ∂^2/∂z^2

2.5 全部足す

3. まとめ

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1. １階微分の極座標表示

2. ラプラシアンΔの極座標表示

2.1 ∂^2/∂x^2

1行目

間違えやすいポイント

2行目

3行目

2.2 ∂^2/∂y^2

1行目

2行目

3行目

2.3 ∂^2/∂z^2

1行目

2行目

2.4 ∂^2/∂x^2 + ∂^2/∂y^2 + ∂^2/∂z^2

2.5 全部足す

3. まとめ

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