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:We find here the generating function of the Legendre polynomials ::Use of the spherical harmonic addition theorem :gives the desired result.
Wikimedia Foundation.
2010.
Look at other dictionaries:
Laplace expansion — This article is about the expansion of the determinant of a square matrix as a weighted sum of determinants of sub matrices. For the expansion of an 1/r potential using spherical harmonical functions, see Laplace expansion (potential). In linear… … Wikipedia
Pierre-Simon Laplace — Laplace redirects here. For the city in Louisiana, see LaPlace, Louisiana. For the joint NASA ESA space mission, see Europa Jupiter System Mission. Pierre Simon, marquis de Laplace Pierre Simon Laplace (1749–1827). Posthumous portrait … Wikipedia
Multipole expansion — A multipole expansion is a mathematical series representing a function that depends on angles usually the two angles on a sphere. These series are useful because they can often be truncated, meaning that only the first few terms need to be… … Wikipedia
List of mathematics articles (L) — NOTOC L L (complexity) L BFGS L² cohomology L function L game L notation L system L theory L Analyse des Infiniment Petits pour l Intelligence des Lignes Courbes L Hôpital s rule L(R) La Géométrie Labeled graph Labelled enumeration theorem Lack… … Wikipedia
Spherical multipole moments — are the coefficients in a series expansionof a potential that varies inversely with the distance R to a source, i.e., as frac{1}{R}. Examples of such potentials are the electric potential, the magnetic potential and the gravitational… … Wikipedia
Multipole moment — In mathematics, especially as applied to physics, multipole moments are the coefficients of a series expansion of a potential due to continuous or discrete sources (e.g., an electric charge distribution). A multipole moment usually involves… … Wikipedia
Dipole — This article is about the electromagnetic phenomenon. For other uses, see dipole (disambiguation). The Earth s magnetic field, approximated as a magnetic dipole. However, the N and S (north and south) poles are labeled here geographically, which… … Wikipedia
fluid mechanics — an applied science dealing with the basic principles of gaseous and liquid matter. Cf. fluid dynamics. [1940 45] * * * Study of the effects of forces and energy on liquids and gases. One branch of the field, hydrostatics, deals with fluids at… … Universalium
geoid — geoidal, adj. /jee oyd/, n. 1. an imaginary surface that coincides with mean sea level in the ocean and its extension through the continents. 2. the geometric figure formed by this surface, an ellipsoid flattened at the poles. [1880 85; < Gk… … Universalium
Legendre polynomials — Note: People sometimes refer to the more general associated Legendre polynomials as simply Legendre polynomials . In mathematics, Legendre functions are solutions to Legendre s differential equation::{d over dx} left [ (1 x^2) {d over dx} P n(x)… … Wikipedia
