Laplace expansion (potential)

Laplace expansion (potential)

:"See also Laplace expansion of determinant".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} = frac{1}{sqrt{r^2 + (r')^2 - 2 r r' cosgamma = frac{1}{r_scriptscriptstyle> sqrt{1 + h^2 - 2 h cosgamma quadhbox{with}quad h equiv frac{r_scriptscriptstyle<}{r_scriptscriptstyle>} . We find here the generating function of the Legendre polynomials P_ell(cosgamma) ::frac{1}{sqrt{1 + h^2 - 2 h cosgamma = sum_{ell=0}^infty h^ell P_ell(cosgamma).Use of the spherical harmonic addition theorem :P_{ell}(cos gamma) = frac{4pi}{2ell + 1} sum_{m=-ell}^{ell} Y^{-m}_{ell}( heta, varphi) Y^m_{ell}( heta', varphi')gives the desired result.


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