中心力場内の粒子の運動/メモ の変更点

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[[量子力学Ⅰ/中心力場内の粒子]]

* 演習:偏微分の計算 [#j884fda6]

** 解答 [#f8689b48]

(1)

&math(r^2=x^2+y^2+z^2); より、&math(2r\frac{\PD r}{\PD x}=2x); などが得られて、

 &math(
\begin{cases}
\displaystyle\frac{\PD r}{\PD x}=\frac{x}{r}=\sin\theta\cos\phi\\[4mm]
\displaystyle\frac{\PD r}{\PD y}=\frac{y}{r}=\sin\theta\sin\phi\\[4mm]
\displaystyle\frac{\PD r}{\PD z}=\frac{z}{r}=\cos\theta\\
\end{cases}
);

(2)

&math(\tan^2\theta=\frac{x^2+y^2}{z^2}); より、
&math(\frac{1}{\cos^2\theta}\frac{\PD \theta}{\PD y}=\frac{2x}{z^2});

&math(\frac{\not\!2\tan\theta}{\cos^2\theta}\frac{\PD \theta}{\PD x}=\frac{\not\! 2x}{z^2});、
&math(\frac{\not\!2\tan\theta}{\cos^2\theta}\frac{\PD \theta}{\PD y}=\frac{\not\! 2y}{z^2});、
&math(\frac{\not\!2\tan\theta}{\cos^2\theta}\frac{\PD \theta}{\PD z}=-\not\!2\frac{x^2+y^2}{z^3});、

 &math(
\begin{cases}
\displaystyle\frac{\PD \theta}{\PD x}=\frac{r\sin\theta\cos\phi}{r^2\cos^2\theta}\frac{\cos^2\theta}{\tan\theta}=\frac{1}{r}\cos\theta\cos\phi\\[4mm]
\displaystyle\frac{\PD \theta}{\PD y}=\frac{r\sin\theta\sin\phi}{r^2\cos^2\theta}\frac{\cos^2\theta}{\tan\theta}=\frac{1}{r}\cos\theta\sin\phi\\[4mm]
\displaystyle\frac{\PD \theta}{\PD z}=-\frac{r^2\sin^2\theta}{r^3\cos^3\theta}\frac{\cos^2\theta}{\tan\theta}=-\frac{1}{r}\sin\theta
\end{cases}
);

(3)

&math(\tan\phi=\frac{y}{x}); より、

&math(\frac{1}{\cos^2\phi}\frac{\PD\phi}{\PD x}=-\frac{y}{x^2});、
&math(\frac{1}{\cos^2\phi}\frac{\PD\phi}{\PD y}=\frac{1}{x});、
&math(\frac{1}{\cos^2\phi}\frac{\PD\phi}{\PD z}=0); であるから、

 &math(
\begin{cases}
\displaystyle\frac{\PD \phi}{\PD x}=-\frac{r\sin\theta\sin\phi}{r^2\sin^2\theta\cos^2\phi}\cos^2\phi=-\frac{\sin\phi}{r\sin\theta}\\[4mm]
\displaystyle\frac{\PD \phi}{\PD y}=\frac{1}{r\sin\theta\cos\phi}\cos^2\phi=\frac{\cos\phi}{r\sin\theta}\\[4mm]
\displaystyle\frac{\PD \phi}{\PD z}=0
\end{cases}
);

* 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(本気?)。

&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}
);

恐らくもっと簡単に求める方法もあるはず。

** 球座標の角運動量演算子 [#nad66919]

&math(
\hat l_x&=-i\hbar\Big(y\frac{\PD}{\PD z}-z\frac{\PD}{\PD y}\Big)\\
&=-i\hbar\bigg[r\sin\theta\sin\phi\Big(\cancel{\cos\theta \frac{\PD}{\PD r}}-\frac{1}{r}\sin\theta \frac{\PD}{\PD \theta}\Big)
-r\cos\theta\Big(\cancel{\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)\bigg]\\
&=i\hbar\Big(\sin\phi\frac{\PD}{\PD\theta}+\frac{\cos\phi}{\tan\theta}\frac{\PD}{\PD\phi}\Big)
);

&math(
\hat l_y&=-i\hbar\Big(z\frac{\PD}{\PD x}-x\frac{\PD}{\PD z}\Big)\\
&=-i\hbar\bigg[r\cos\theta\Big(\cancel{\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)
-r\sin\theta\cos\phi\Big(\cancel{\cos\theta \frac{\PD}{\PD r}}
-\frac{1}{r}\sin\theta \frac{\PD}{\PD \theta}\Big)\bigg]\\
&=i\hbar\Big(-\cos\phi\frac{\PD}{\PD\theta}+\frac{\sin\phi}{\tan\theta}\frac{\PD}{\PD\phi}\Big)
);

&math(
\hat l_z&=-i\hbar\Big(x\frac{\PD}{\PD y}-y\frac{\PD}{\PD x}\Big)\\
&=-i\hbar\bigg[r\sin\theta\cos\phi\Big(\cancel{\sin\theta\sin\phi \frac{\PD}{\PD r}}
+\cancel{\frac{1}{r}\cos\theta\sin\phi \frac{\PD}{\PD \theta}}
+\frac{\cos\phi}{r\sin\theta} \frac{\PD}{\PD \phi}\Big)\\
&\hspace{1cm}-r\sin\theta\sin\phi\Big(\cancel{\sin\theta\cos\phi \frac{\PD}{\PD r}}
+\cancel{\frac{1}{r}\cos\theta\cos\phi \frac{\PD}{\PD \theta}}
-\frac{\sin\phi}{r\sin\theta} \frac{\PD}{\PD \phi}\Big)\bigg]\\
&=-i\hbar\frac{\PD}{\PD\phi}
);

&math(
\hat{\bm l}^2&=\hat l_x^2+\hat l_y^2+\hat l_z^2\\
&=
-\hbar^2\Big(\sin\phi\frac{\PD}{\PD\theta}+\frac{\cos\phi}{\tan\theta}\frac{\PD}{\PD\phi}\Big)^2
-\hbar^2\Big(-\cos\phi\frac{\PD}{\PD\theta}+\frac{\sin\phi}{\tan\theta}\frac{\PD}{\PD\phi}\Big)^2
-\hbar^2\frac{\PD^2}{\PD\phi^2}\\
&=
-\hbar^2\Big(
\sin^2\phi\frac{\PD^2}{\PD\theta^2}
-\cancel{\frac{\sin\phi\cos\phi}{\sin^2\theta}\frac{\PD}{\PD\phi}}
+\cancel{\frac{2\sin\phi\cos\phi}{\tan\theta}\frac{\PD}{\PD\theta}\frac{\PD}{\PD\phi}}
+\frac{\cos^2\phi}{\tan\theta}\frac{\PD}{\PD\theta}
+\cancel{\frac{-\sin\phi\cos\phi}{\tan^2\theta}\frac{\PD}{\PD\phi}}
+\frac{\cos^2\phi}{\tan^2\theta}\frac{\PD^2}{\PD\phi^2}
\Big)\\
&\ \ \ -\hbar^2\Big(
\cos^2\phi\frac{\PD^2}{\PD^2\theta}
+\cancel{\frac{\sin\phi\cos\phi}{\sin^2\theta}\frac{\PD}{\PD\phi}}
-\cancel{\frac{2\sin\phi\cos\phi}{\tan\theta}\frac{\PD}{\PD\theta}\frac{\PD}{\PD\phi}}
+\frac{\sin^2\phi}{\tan\theta}\frac{\PD}{\PD\theta}
+\cancel{\frac{\sin\phi\cos\phi}{\tan^2\theta}\frac{\PD}{\PD\phi}}
+\frac{\sin^2\phi}{\tan^2\theta}\frac{\PD^2}{\PD\phi^2}
\Big)\\
&\ \ \ -\hbar^2\frac{\PD^2}{\PD\phi^2}\\
&=-\hbar^2\Big[\frac{\PD^2}{\PD\theta^2}+\frac{1}{\tan\theta}\frac{\PD}{\PD\theta}+\Big(\frac{1}{\tan^2\theta}+1\Big)\frac{\PD^2}{\PD\phi^2}\Big]\\
&=-\hbar^2\Big(\frac{\PD^2}{\PD\theta^2}+\frac{\cos\theta}{\sin\theta}\frac{\PD}{\PD\theta}+\frac{1}{\sin^2\theta}\frac{\PD^2}{\PD\phi^2}\Big)\\
&=-\hbar^2\Big[\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}\Big]\\
&=-\hbar^2\hat\Lambda
);

** 球関数 $Y^m_l(\theta,\phi)$:角運動量の固有関数 [#s564caed]
&mathjax;

 LANG:mathematica
 Table[
   Table[
     ParametricPlot3D[
       Module[{r = Abs[SphericalHarmonicY[l, m, theta, phi]]^2},
         {r Sin[theta] Cos[phi], r Sin[theta] Sin[phi], r Cos[theta]}
       ], {theta, 0, Pi}, {phi, 0, 2 Pi}, 
       PlotPoints -> 100, ImageSize -> Large,
       PlotRange -> {{-0.20, 0.20}, {-0.20, 0.20}, {-0.4, 0.4}}
     ],
     {m, 0, l}
   ],
   {l, 0, 4}
 ]

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