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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.
Additional recommended knowledge
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
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,
The derivation of this expansion is simple. One writes
We find here the generating function of the Legendre polynomials :
Use of the spherical harmonic addition theorem
gives the desired result.
|This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Laplace_expansion_(potential)". A list of authors is available in Wikipedia.|