中心力場内の粒子の運動/メモ のバックアップ(No.2)

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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

解答

(1)

r^2=x^2+y^2+z^2 より、 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)

\tan^2\theta=\frac{x^2+y^2}{z^2} より、 \frac{1}{\cos^2\theta}\frac{\PD \theta}{\PD y}=\frac{2x}{z^2}

\frac{\not\!2\tan\theta}{\cos^2\theta}\frac{\PD \theta}{\PD x}=\frac{\not\! 2x}{z^2} \frac{\not\!2\tan\theta}{\cos^2\theta}\frac{\PD \theta}{\PD y}=\frac{\not\! 2y}{z^2} \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)

\tan\phi=\frac{y}{x} より、

\frac{1}{\cos^2\phi}\frac{\PD\phi}{\PD x}=-\frac{y}{x^2} \frac{1}{\cos^2\phi}\frac{\PD\phi}{\PD y}=\frac{1}{x} \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

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

  1. \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}

  1. \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}
  1. \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}
  1. \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}
  1. \frac{\sin\theta\cos\theta\cos^2\phi}{r} \frac{\PD}{\PD r}\frac{\PD}{\PD \theta}
  2. \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}\\ &\ \ \
  1. \frac{\cos^2\theta\cos^2\phi}{r} \frac{\PD}{\PD r}
  2. \frac{\sin\theta\cos\theta\cos^2\phi}{r} \frac{\PD}{\PD r}\frac{\PD}{\PD \theta}
  3. \frac{-\sin\theta\cos\theta\cos^2\phi}{r^2}\frac{\PD}{\PD \theta}
  4. \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} \\ &\ \ \
  1. \frac{\sin^2\phi}{r} \frac{\PD}{\PD r}
  • \frac{\sin\phi\cos\phi}{r} \frac{\PD}{\PD r}\frac{\PD}{\PD \phi}
  1. \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}
  1. \frac{\sin\phi\cos\phi}{r^2\sin^2\theta} \frac{\PD}{\PD \phi}
  2. \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}
  1. \frac{2\sin\theta\cos\theta\cos^2\phi}{r} \frac{\PD}{\PD r}\frac{\PD}{\PD \theta}
  2. \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}\\ &\ \ \
  1. \frac{\cos^2\theta\cos^2\phi}{r} \frac{\PD}{\PD r}
  2. \frac{-\sin\theta\cos\theta\cos^2\phi}{r^2}\frac{\PD}{\PD \theta}
  3. \frac{\cos^2\theta\cos^2\phi}{r^2} \frac{\PD^2}{\PD \theta^2}
  4. \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}
  1. \frac{\sin^2\phi}{r} \frac{\PD}{\PD r}
  2. \frac{\cos\theta\sin^2\phi}{r^2\sin\theta} \frac{\PD}{\PD \theta}
  3. \frac{\sin\phi\cos\phi}{r^2\sin^2\theta} \frac{\PD}{\PD \phi}
  4. \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}

  1. \frac{1}{r}\cos\theta\sin\phi \frac{\PD}{\PD \theta}
  2. \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}
  1. \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}
  1. \frac{2\sin\phi\cos\phi}{r} \frac{\PD}{\PD r}\frac{\PD}{\PD \phi}\\ &\ \ \
  2. \frac{\cos^2\theta\sin^2\phi}{r} \frac{\PD}{\PD r}
  3. \frac{-\sin\theta\cos\theta\sin^2\phi}{r^2} \frac{\PD}{\PD \theta}
  4. \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}
  1. \frac{2\cos\theta\sin\phi\cos\phi}{r^2\sin\theta} \frac{\PD}{\PD \theta}\frac{\PD}{\PD \phi}\\ &\ \ \
  2. \frac{\cos^2\phi}{r} \frac{\PD}{\PD r}
  3. \frac{\cos\theta\cos^2\phi}{r^2\sin\theta} \frac{\PD}{\PD \theta}
  4. \frac{-\sin\phi\cos\phi}{r^2\sin^2\theta} \frac{\PD}{\PD \phi}
  5. \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}
  1. \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} \\ &\ \ \
  1. \frac{\sin^2\theta}{r} \frac{\PD}{\PD r}
  2. \frac{\sin\theta\cos\theta}{r^2} \frac{\PD}{\PD \theta}
  3. \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}
  1. \frac{2\sin\theta\cos\theta\cos^2\phi}{r} \frac{\PD}{\PD r}\frac{\PD}{\PD \theta}
  2. \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}\\ &\ \ \
  1. \frac{\cos^2\theta\cos^2\phi}{r} \frac{\PD}{\PD r}
  2. \frac{-\sin\theta\cos\theta\cos^2\phi}{r^2}\frac{\PD}{\PD \theta}
  3. \frac{\cos^2\theta\cos^2\phi}{r^2} \frac{\PD^2}{\PD \theta^2}
  4. \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}
  1. \frac{\sin^2\phi}{r} \frac{\PD}{\PD r}
  2. \frac{\cos\theta\sin^2\phi}{r^2\sin\theta} \frac{\PD}{\PD \theta}
  3. \frac{\sin\phi\cos\phi}{r^2\sin^2\theta} \frac{\PD}{\PD \phi}
  4. \frac{\sin^2\phi}{r^2\sin^2\theta} \frac{\PD^2}{\PD \phi^2}\\

&\ \ \

  1. \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}
  1. \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}
  1. \frac{2\sin\phi\cos\phi}{r} \frac{\PD}{\PD r}\frac{\PD}{\PD \phi}\\ &\ \ \
  2. \frac{\cos^2\theta\sin^2\phi}{r} \frac{\PD}{\PD r}
  3. \frac{-\sin\theta\cos\theta\sin^2\phi}{r^2} \frac{\PD}{\PD \theta}
  4. \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}
  1. \frac{2\cos\theta\sin\phi\cos\phi}{r^2\sin\theta} \frac{\PD}{\PD \theta}\frac{\PD}{\PD \phi}\\ &\ \ \
  2. \frac{\cos^2\phi}{r} \frac{\PD}{\PD r}
  3. \frac{\cos\theta\cos^2\phi}{r^2\sin\theta} \frac{\PD}{\PD \theta}
  4. \frac{-\sin\phi\cos\phi}{r^2\sin^2\theta} \frac{\PD}{\PD \phi}
  5. \frac{\cos^2\phi}{r^2\sin^2\theta} \frac{\PD^2}{\PD \phi^2}\\

&\ \ \

  1. \cos^2\theta \frac{\PD^2}{\PD r^2}
  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} \\&\ \ \
  1. \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} \\&\ \ \
  2. \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}
  3. \cancel{\frac{\sin\theta\cos\theta}{r^2}}+\cancel{\frac{\sin\theta\cos\theta}{r^2}}\Big) \frac{\PD}{\PD \theta} \\&\ \ \
  4. \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} \\&\ \ \
  5. \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} \\&\ \ \
  6. \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} \\&\ \ \
  7. \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} \\&\ \ \
  8. \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} \\&\ \ \
  9. \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}

  1. \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}
  2. \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} );

球座標の角運動量演算子

&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}}
  1. \frac{1}{r}\cos\theta\sin\phi \frac{\PD}{\PD \theta}
  2. \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}}

  1. \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}}

  1. \cancel{\frac{1}{r}\cos\theta\sin\phi \frac{\PD}{\PD \theta}}
  2. \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}}
  3. \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}}
  1. \cancel{\frac{2\sin\phi\cos\phi}{\tan\theta}\frac{\PD}{\PD\theta}\frac{\PD}{\PD\phi}}
  2. \frac{\cos^2\phi}{\tan\theta}\frac{\PD}{\PD\theta}
  3. \cancel{\frac{-\sin\phi\cos\phi}{\tan^2\theta}\frac{\PD}{\PD\phi}}
  4. \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}
  5. \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}}
  1. \frac{\sin^2\phi}{\tan\theta}\frac{\PD}{\PD\theta}
  2. \cancel{\frac{\sin\phi\cos\phi}{\tan^2\theta}\frac{\PD}{\PD\phi}}
  3. \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

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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