三次元空間での散乱現象/メモ のバックアップ(No.1)

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&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}

1. \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
2. \frac{k_0\cos\theta}{r}e^{-ik_0r(1-\cos\theta)}f^*(\theta,\phi)
3. \left(-\frac{1}{ir^2}+\frac{k_0}{r}\right)e^{ik_0r(1-\cos\theta)}f(\theta,\phi) \right]\\ &=k_0\cos\theta+\frac{k_0}{r^2}|f(\theta,\phi)|^2
4. \frac{\hbar}{m}\left(\frac{k_0(1+\cos\theta)}{r}\mathrm{Re}Z-\frac{1}{r^2}\mathrm{Im}Z\right) );

ただし、