Laplace Expansion (potential)

Laplace Expansion (potential)

In physics, the Laplace expansion of a 1/r - type potential is applied to expand Newton's gravitational potential or Coulomb's electrostatic potential. In quantum mechanical calculations on atoms the expansion is used in the evaluation of integrals of the interelectronic repulsion.

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

\frac{1}{|\mathbf{r}-\mathbf{r}'|} = \sum_{\ell=0}^\infty \frac{4\pi}{2\ell+1} \sum_{m=-\ell}^{\ell} (-1)^m \frac{r_{{\scriptscriptstyle<}}^\ell }{r_{{\scriptscriptstyle>}}^{\ell+1} } Y^{-m}_\ell(\theta, \varphi) Y^{m}_\ell(\theta', \varphi').

Here r has the spherical polar coordinates (r, θ, φ) and r' has ( r', θ', φ'). Further r< is min(r, r') and r> is max(r, r'). The function is a normalized spherical harmonic function. The expansion takes a simpler form when written in terms of solid harmonics,

\frac{1}{|\mathbf{r}-\mathbf{r}'|} = \sum_{\ell=0}^\infty \sum_{m=-\ell}^{\ell} (-1)^m I^{-m}_\ell(\mathbf{r}) R^{m}_\ell(\mathbf{r}')\quad\hbox{with}\quad |\mathbf{r}| > |\mathbf{r}'|.

Read more about Laplace Expansion (potential):  Derivation

Famous quotes containing the words laplace and/or expansion:

    “Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective positions of the beings which compose it, if moreover this intelligence were vast enough to submit these data to analysis, it would embrace in the same formula both the movements of the largest bodies in the universe and those of the lightest atom; to it nothing would be uncertain, and the future as the past would be present to its eyes.”
    —Pierre Simon De Laplace (1749–1827)

    “Every expansion of government in business means that government in order to protect itself from the political consequences of its errors and wrongs is driven irresistibly without peace to greater and greater control of the nation’s press and platform. Free speech does not live many hours after free industry and free commerce die.”
    —Herbert Hoover (1874–1964)