Hartree equation

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In 1927, a year after the publication of the Schrödinger equation, Hartree formulated what are now known as the Hartree equations for atoms, using the concept of self-consistency that Lindsay had introduced in his study of many electron systems in the context of Bohr theory. Hartree assumed that the nucleus together with the electrons formed a spherically symmetric field. The charge distribution of each electron was the solution of the Schrödinger equation for an electron in a potential  v(r) , derived from the field. Self-consistency required that the final field, computed from the solutions was self-consistent with the initial field and he called his method the Self-consistent Field method.

In order to solve the equation of an electron in a spherical potential, Hartree first introduced atomic units to eliminate physical constants. Then he converted the Laplacian from Cartesian to spherical coordinates to show that the solution was a product of a radial function  P(r)/r and a spherical harmonic with an angular quantum number \ell, namely \psi=(1/r)P(r)S_\ell(\theta,\phi). The equation for the radial function was

d^2P(r)/dr^2 + [2(E-v(r)) - \ell(\ell+1)/r^2]P(r)=0.

In mathematics, the Hartree equation, named after Douglas Hartree, is

i\,\partial_tu + \nabla^2 u= V(u)u

in \mathbb{R}^{d+1} where

V(u)= \pm |x|^{-n} * |u|^2


 0 < n < d

The non-linear Schrödinger equation is in some sense a limiting case.

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