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| :''See also [[Laplace expansion|Laplace expansion of determinant]]''.
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| In physics, the '''Laplace expansion''' of a 1/''r'' - type potential is applied to expand [[ Newton's law of universal gravitation#Gravitational field|Newton's gravitational potential]] or [[Coulomb's law#Table of derived quantities|Coulomb's electrostatic potential]]. In quantum mechanical calculations on atoms the expansion is used in the evaluation of integrals of the interelectronic repulsion.
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| The Laplace expansion is in fact the expansion of the inverse distance between two points. Let the points have position vectors '''r''' and '''r'''', then the Laplace expansion is
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| :<math>
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| \frac{1}{|\mathbf{r}-\mathbf{r}'|} = \sum_{\ell=0}^\infty \frac{4\pi}{2\ell+1}
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| \sum_{m=-\ell}^{\ell}
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| (-1)^m \frac{r_{{\scriptscriptstyle<}}^\ell }{r_{{\scriptscriptstyle>}}^{\ell+1} } Y^{-m}_\ell(\theta, \varphi) Y^{m}_\ell(\theta', \varphi').
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| </math>
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| Here '''r''' has the spherical polar coordinates (''r'', θ, φ) and '''r''''
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| has ( ''r''', θ', φ').
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| Further ''r''<sub><</sub>
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| is min(''r'', ''r''') | |
| and ''r''<sub>></sub> is max(''r'', ''r''').
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| The function <math>Y^m_{\ell}</math> is a normalized [[spherical harmonics|spherical harmonic function]]. The expansion takes a simpler form when written in terms of [[solid harmonics]],
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| :<math> | |
| \frac{1}{|\mathbf{r}-\mathbf{r}'|} = \sum_{\ell=0}^\infty
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| \sum_{m=-\ell}^{\ell}
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| (-1)^m I^{-m}_\ell(\mathbf{r}) R^{m}_\ell(\mathbf{r}')\quad\hbox{with}\quad |\mathbf{r}| > |\mathbf{r}'|. | |
| </math>
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| ==Derivation==
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| One writes
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| :<math> | |
| \frac{1}{|\mathbf{r}-\mathbf{r}'|} = \frac{1}{\sqrt{r^2 + (r')^2 - 2 r r' \cos\gamma}} =
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| \frac{1}{r_{{\scriptscriptstyle>}} \sqrt{1 + h^2 - 2 h \cos\gamma}} \quad\hbox{with}\quad h \equiv \frac{r_{{\scriptscriptstyle<}}}{r_{{\scriptscriptstyle>}}} .
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| </math>
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| We find here the generating function of the [[Legendre polynomials#Applications of Legendre polynomials in physics|Legendre polynomials]] <math>P_\ell(\cos\gamma)</math> :
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| :<math>
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| \frac{1}{\sqrt{1 + h^2 - 2 h \cos\gamma}} = \sum_{\ell=0}^\infty h^\ell P_\ell(\cos\gamma).
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| </math>
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| Use of the [[Spherical multipole moments#Spherical multipole moments of a point charge|spherical harmonic addition theorem]]
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| :<math>
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| P_{\ell}(\cos \gamma) = \frac{4\pi}{2\ell + 1} \sum_{m=-\ell}^{\ell}
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| (-1)^m Y^{-m}_{\ell}(\theta, \varphi) Y^m_{\ell}(\theta', \varphi')
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| </math>
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| gives the desired result.
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| [[Category:Potential theory]]
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| [[Category:Atomic physics]]
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| [[Category:Rotational symmetry]]
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