Self-consistent field approximation and uniform field approximation? Can anyone give me explanation of self-consistent field approximation and uniform field approximation?  I know self-consistent as when we write the Schrödinger equation as
$$[ -\frac{\hbar^2}{2m} \nabla^2 +v(r) ] \psi{_k}(r)~=~ E(k)\psi{_k}(r).$$
To know the potential appearing in the above equation, we have to find its solution.
If we consider a solution for determining potential and get the same value of potential by considering another solution then this is called self-consistency. I'm not actually clear on this two topics. Can anyone point me in the right direction?
 A: ad "self-consistent field theory": 
In quantum many-body physics self-consistent field theory or mean field theory (MFT) is a method to approximate the (ground state) wave function (and energy) of an interacting many body system. (the Hamiltonian above looks like a single-particle Hamiltonian; for a "true" interacting many-body Hamiltonian see below). 
$$    
H=\sum_{i}\frac{p_{i}^{2}}{2m}+V_{sp}\left(x_{i}\right)+\frac{1}{2}\sum_{i\neq j}V_{int}\left(x_{i}-x_{j}\right)
$$
The idea of MFT is to approximately incorporate the particle-particle interactions by modifying the single particle potential $V_{sp}\longrightarrow \bar{V}_{sp}$. Thus one is left with an effective single particle problem which is linear. The details on how to correctly change the single particle potential can be found in http://www.physics.metu.edu.tr/~hande/teaching/741-lectures/lecture-04.pdf (in the context of theoretical chemistry), in http://en.wikipedia.org/wiki/Mean_field_theory#Formal_approach (in the context of statistical mechanics) and in http://www.phys.lsu.edu/~jarrell/COURSES/ADV_SOLID_HTML/Other_online_texts/Many-body%20quantum%20theory%20in%0Acondensed%20matter%20physics%0AHenrik%20Bruus%20and%20Karsten%20Flensberg.pdf (on p.81 in the much more suited language of 2nd quantization).
ad "uniform field approximation": I have never come across this term but maybe the following is what you meant: approximation a particle in a periodic potential by a particle in a uniform constant potential (see p.35 of http://www.phys.lsu.edu/~jarrell/COURSES/ADV_SOLID_HTML/Other_online_texts/Many-body%20quantum%20theory%20in%0Acondensed%20matter%20physics%0AHenrik%20Bruus%20and%20Karsten%20Flensberg.pdf); this is also known as "Jellium model". Low energy electrons "see" a smeared-out (=constant) ionic potential background (see below). 

(from "Many Body Quantum Thoery in condensed matter physics"; Flensberg, Bruus )
