中心力場内の粒子の運動/メモ のバックアップの現在との差分(No.1)
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[[量子力学Ⅰ/中心力場内の粒子]] [[量子力学Ⅰ/中心力場内の粒子の運動]] * 球座標表示のラプラシアン [#o621468a] ** 球関数 $Y^m_l(\theta,\phi)$:角運動量の固有関数 [#s564caed] &katex(); &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\\ 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} ] &=\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。 &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} );
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