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A further simplification of the Hartree equations, Eq. (IV.1.3), is achieved by replacing the Hartree potential by its angular average,
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|
|
(1.5) |
This still depends on all the wave functions
, but as the one-particle potential now is spherically symmetric, we can use the decomposition into spherical harmonics, radial wave functions, and spin,
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(1.6) |
Here, the index
indicates that we are back to our usual quantum numbers
that we know from the hydrogen atom. In contrast to the latter, the radial functions now depend on
and
because we do not have the simple
Coulomb potential as one-particle potential.
An even cruder approximation to
would be a parametrization of the form
by which one loses the self-consistency and ends up with one single Schrödinger equation for a particle in the potential
.
Exercise: Give a physical argument for the condition
in the above equation.
Subsections
Next: Periodic Table
Up: The Hartree Equations, Atoms,
Previous: Effective Average Potential
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Tobias Brandes
2005-04-26