# 球多極矩

## 點電荷案例

$\Phi(\mathbf{r}) = \frac{q}{4\pi\varepsilon_0|\mathbf{r} - \mathbf{r^{\prime}}|}= \frac{q}{4\pi\varepsilon_0}\frac{1}{\sqrt{r^{2} + r^{\prime 2} - 2 r^{\prime} r \cos \gamma}}$

$\Phi(\mathbf{r}) = \frac{q}{4\pi\varepsilon_0 r} \sum_{\ell=0}^{\infty} \left( \frac{r^{\prime}}{r} \right)^{\ell} P_{\ell}(\cos \gamma )$

$\cos \gamma = \cos \theta \cos \theta^{\prime} + \sin \theta \sin \theta^{\prime} \cos(\phi - \phi^{\prime})$

$P_{\ell}(\cos \gamma) = \frac{4\pi}{2\ell + 1} \sum_{m=-\ell}^{\ell} Y_{\ell m}(\theta, \phi) Y_{\ell m}^{*}(\theta^{\prime}, \phi^{\prime})$

$\Phi(\mathbf{r}) = \frac{q}{4\pi\varepsilon_0 r} \sum_{\ell=0}^{\infty}\left( \frac{r^{\prime}}{r} \right)^{\ell}\left( \frac{4\pi}{2\ell+1} \right)\sum_{m=-\ell}^{\ell} Y_{\ell m}(\theta, \phi) Y_{\ell m}^{*}(\theta^{\prime}, \phi^{\prime})$

$q_{\ell m} \ \stackrel{\mathrm{def}}{=}\ q r^{\prime\ell} Y_{\ell m}^{*}(\theta^{\prime}, \phi^{\prime})$

$\Phi(\mathbf{r}) = \frac{1}{\varepsilon_0} \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} \frac{q_{\ell m} Y_{\ell m}(\theta, \phi)}{(2\ell+1)r^{\ell+1}}$

$\Phi(\mathbf{r}) = \frac{q}{4\pi\varepsilon_0 r^{\prime}} \sum_{\ell=0}^{\infty}\left( \frac{r}{r^{\prime}} \right)^{\ell}\left( \frac{4\pi}{2\ell+1} \right)\sum_{m=-\ell}^{\ell} Y_{\ell m}(\theta, \phi) Y_{\ell m}^{*}(\theta^{\prime}, \phi^{\prime})$

$I_{\ell m} \ \stackrel{\mathrm{def}}{=}\ \frac{q}{\left( r^{\prime} \right)^{\ell+1}} Y_{\ell m}^{*}(\theta^{\prime}, \phi^{\prime})$

$\Phi(\mathbf{r}) = \frac{1}{\varepsilon_0} \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} \frac{ I_{\ell m} r^{\ell}Y_{\ell m}(\theta, \phi) }{2\ell+1}$

## 電荷密度案例

$\Phi(\mathbf{r}) = \frac{1}{\varepsilon_0} \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} \frac{q_{\ell m} Y_{\ell m}(\theta, \phi)}{(2\ell+1)r^{\ell+1}}$

### 內部球多極矩

$\Phi(\mathbf{r}) = \frac{1}{\varepsilon_0} \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} \frac{ I_{\ell m} r^{\ell}Y_{\ell m}(\theta, \phi) }{2\ell+1}$

## 兩個球多極矩之間的相互作用能

$U = \int_{\mathbb{V}} \rho_{2}(\mathbf{r}) \Phi_{1}(\mathbf{r})\ \mathrm{d}^3\mathbf{r}$

$\Phi_1(\mathbf{r}) = \frac{1}{\varepsilon_0} \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} \frac{q_{1\ell m} Y_{\ell m}(\theta, \phi)}{(2\ell+1)r^{\ell+1}}$

$U = \frac{1}{\varepsilon_0} \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} \frac{q_{1\ell m}}{2\ell+1}\int_{\mathbb{V}} \frac{\rho_{2}(\mathbf{r})Y_{\ell m}(\theta, \phi)}{r^{\ell+1}} \ \mathrm{d}^3\mathbf{r}$

$U = \frac{1}{\varepsilon_0} \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} \frac{q_{1\ell m} I_{2\ell m}^{*}}{2\ell+1}$

## 軸對稱特別案例

$P_{\ell}(\cos \theta) =\sqrt{\frac{4\pi}{2\ell+1}} Y_{\ell0}(\theta, \phi)$

$q_{\ell} \ \stackrel{\mathrm{def}}{=}\ \int_{\mathbb{V}'} \sqrt{\frac{2\ell+1}{4\pi}}\ \rho(\mathbf{r}^{\prime}) \left( r^{\prime} \right)^{\ell} P_{\ell}(\cos \theta^{\prime})\ \mathrm{d}^3\mathbf{r}^{\prime}$

$\Phi(\mathbf{r}) = \frac{1}{4\pi\varepsilon_0} \sum_{\ell=0}^{\infty}\sqrt{\frac{4\pi}{2\ell+1}}\ \frac{q_{\ell} P_{\ell}(\cos \theta)}{r^{\ell+1}}$

$I_{\ell} \ \stackrel{\mathrm{def}}{=}\ \int_{\mathbb{V}'} \sqrt{\frac{2\ell+1}{4\pi}}\ \frac{\rho(\mathbf{r}^{\prime})}{\left( r^{\prime} \right)^{\ell+1}}P_{\ell}(\cos \theta^{\prime})\ \mathrm{d}^3\mathbf{r}^{\prime}$

$\Phi(\mathbf{r}) = \frac{1}{4\pi\varepsilon_0} \sum_{\ell=0}^{\infty} \sqrt{\frac{4\pi}{2\ell+1}}I_{\ell} r^{\ell} P_{\ell}(\cos \theta)$

## 球多極矩的表達式

\begin{align} q_{00} & =\frac{1}{\sqrt{4\pi}}\int_{\mathbb{V'}}\rho(\mathbf{r}')\ \mathrm{d}^3\mathbf{r}' & & =\frac{1}{\sqrt{4\pi}}\ q \\ q_{11} & = -\sqrt{\frac{3}{8\pi}}\int_{\mathbb{V'}} r' \sin{\theta'}\ e^{-i\phi'}\rho(\mathbf{r}')\ \mathrm{d}^3\mathbf{r}' & & = -\sqrt{\frac{3}{8\pi}}\ (p_x - ip_y) \\ q_{10} & =\sqrt{\frac{3}{4\pi}}\int_{\mathbb{V'}} r' \cos{\theta}\ \rho(\mathbf{r}')\ \mathrm{d}^3\mathbf{r}' & & = -\sqrt{\frac{3}{4\pi}}\ p_z \\ q_{22} & =\sqrt{\frac{15}{32\pi}}\int_{\mathbb{V'}} r^{\prime 2} \sin^2{\theta'}\ e^{-2i\phi'}\rho(\mathbf{r}')\ \mathrm{d}^3\mathbf{r}' & & =\sqrt{\frac{15}{288\pi}}\ (Q_{11}-2iQ_{12}-Q_{22}) \\ q_{21} & = - \sqrt{\frac{15}{8\pi}}\int_{\mathbb{V'}} r^{\prime 2} \sin{\theta'}\cos{\theta'}\ e^{-i\phi'}\rho(\mathbf{r}')\ \mathrm{d}^3\mathbf{r}' & & = - \sqrt{\frac{15}{72\pi}}\ (Q_{13}-iQ_{33}) \\ q_{20} & =\sqrt{\frac{5}{16\pi}}\int_{\mathbb{V'}} r^{\prime 2}(\cos^2{\theta'}-1)\rho(\mathbf{r}')\ \mathrm{d}^3\mathbf{r}' & & =\sqrt{\frac{5}{16\pi}}\ Q_{33} \end{align}

## 參考文獻

1. ^ Griffiths, David J., Introduction to Electrodynamics (3rd ed.), Prentice Hall, pp. 146–148, 1998, ISBN 0-13-805326-X
2. ^ 2.0 2.1 2.2 Jackson, John David, Classical Electrodynamic 3rd., USA: John Wiley & Sons, Inc., pp. 107–111, 1999, ISBN 978-0-471-30932-1