Laplace expansion (potential)
Encyclopedia
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
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,
We find here the generating function of the Legendre polynomials
:
Use of the spherical harmonic addition theorem
gives the desired result.
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 functionSpherical harmonics
In mathematics, spherical harmonics are the angular portion of a set of solutions to Laplace's equation. Represented in a system of spherical coordinates, Laplace's spherical harmonics Y_\ell^m are a specific set of spherical harmonics that forms an orthogonal system, first introduced by Pierre...
. The expansion takes a simpler form when written in terms of solid harmonics,
Derivation
The derivation of this expansion is simple. One writesWe find here the generating function of the Legendre polynomials
:Use of the spherical harmonic addition theorem
gives the desired result.