中心力場内の粒子の運動/メモ のバックアップ(No.1)

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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

&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} );

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