Hartree equation

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

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

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

$i\,\partial _{t}u+\nabla ^{2}u=V(u)u$

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

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

and

$0<n<d$

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

External links

• "Hartree equation". Dispersive PDE Wiki.