球多極矩

點電荷案例

${\displaystyle \Phi (\mathbf {r} )={\frac {q}{4\pi \varepsilon _{0}|\mathbf {r} -\mathbf {r^{\prime }} |}}={\frac {q}{4\pi \varepsilon _{0}}}{\frac {1}{\sqrt {r^{2}+r^{\prime 2}-2r^{\prime }r\cos \gamma }}}}$

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

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

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

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

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

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

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

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

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

電荷密度案例

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

內部球多極矩

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

兩個球多極矩之間的相互作用能

${\displaystyle U=\int _{\mathbb {V} }\rho _{2}(\mathbf {r} )\Phi _{1}(\mathbf {r} )\ \mathrm {d} ^{3}\mathbf {r} }$

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

${\displaystyle U={\frac {1}{\varepsilon _{0}}}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }{\frac {q_{1\ell m}}{2\ell +1}}\int _{\mathbb {V} }{\frac {\rho _{2}(\mathbf {r} )Y_{\ell m}(\theta ,\phi )}{r^{\ell +1}}}\ \mathrm {d} ^{3}\mathbf {r} }$

${\displaystyle U={\frac {1}{\varepsilon _{0}}}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }{\frac {q_{1\ell m}I_{2\ell m}^{*}}{2\ell +1}}}$

軸對稱特別案例

${\displaystyle P_{\ell }(\cos \theta )={\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell 0}(\theta ,\phi )}$

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

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

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

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

球多極矩的表達式

{\displaystyle {\begin{aligned}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{aligned}}}

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. Jackson, John David, Classical Electrodynamic 3rd., USA: John Wiley & Sons, Inc.: pp. 107–111, 1999, ISBN 978-0-471-30932-1