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}
);


Counter: 907 (from 2010/06/03), today: 1, yesterday: 1