三次元空間での散乱現象/メモ のバックアップソース(No.3)

更新

* 確率密度の保存 [#m224c56a]

計算途中で検算もまだです。

&math(
S_{r}
&=\mathrm{Re}\left[\varphi^*(r,\theta,\phi)\frac{\hbar}{im}\frac{\PD}{\PD r}\varphi(r,\theta,\phi)\right]\\
&=\mathrm{Re}\left[\frac{\hbar}{im}
\left\{e^{-ik_0r\cos\theta}+\frac{e^{-ik_0r}}{r}f^*(\theta,\phi)\right\}
\left\{ik_0\cos\theta e^{ik_0r\cos\theta}
+\left(-\frac{1}{r^2}+\frac{ik_0}{r}\right)e^{ik_0r}f(\theta,\phi)\right\}
\right]\\
&=\frac{\hbar}{m}\mathrm{Re}\left[k_0\cos\theta+\frac{k_0}{r^2}|f(\theta,\phi)|^2
+\frac{k_0\cos\theta}{r}e^{-ik_0r(1-\cos\theta)}f^*(\theta,\phi)
+\left(-\frac{1}{ir^2}+\frac{k_0}{r}\right)e^{ik_0r(1-\cos\theta)}f(\theta,\phi)
\right]\\
&=\frac{\hbar}{m}\left\{k_0\cos\theta+\frac{k_0}{r^2}|f(\theta,\phi)|^2
+\left(\frac{k_0(1+\cos\theta)}{r}\mathrm{Re}Z-\frac{1}{r^2}\mathrm{Im}Z\right)\right\}
);

ただし、&math(Z=e^{ik_0r(1-\cos\theta)}f(\theta,\phi));

* ラザフォード散乱 [#l9f6a2e8]

 LANG:mathematica
 PolarPlot[
   1/(1 + {0.5, 1, 2, 4}^2 Sin[t/2]^2) // Evaluate, {t, 0, 2 Pi}, 
   PlotLegends -> {"A=0.5", "A=1", "A=2", "A=4"}
 ]
 
 PolarPlot[
   1/(1 + {0.25, 1, 4, 16} Sin[t/2]^2)^2 // Evaluate, {t, 0, 2 Pi}, 
   PlotLegends -> {"A=0.5", "A=1", "A=2", "A=4"}
 ]

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