What potential energy functions are mostly used in Schrodinger equation? In the time-independent Schrodinger equation
$$\left(-\frac{\hbar^2}{2m}\Delta+V(\mathbf{r})\right)\psi(\mathbf{r})=E\psi(\mathbf{r})$$
What functions $V(\mathbf{r})$ are mostly used in research (e.g. Constant potential, Inverse Power-Law Potentials, Finite square well and Infinite square well)? 
And who can give a brief history of different potential functions studied in Schrodinger equation?
And can anyone explain to me what is 'Hartree Potential'?
 A: The most general widely studied examples of potentials are those that are derived from an exactly given ground state, and which have the property that they contain enough parameters to be closed under taking supersymmetric conjugates. These are called "shape invariant" potentials.
Given a real positive ground state, one can ask "Which potential has this ground state?" The answer is that if the ground state is $\exp(-W(x))$, and its energy is exactly zero, then the potential is:
$ V(x) = {1\over 2} |\nabla W|^2 - {1\over 2} \nabla^2 W$
The conjugate potential is defined with a plus sign between the two terms instead of a minus sign. It is more usual to define W as the derivative of what I am calling W, but this convention is terrible in higher than one dimension.
The two potentials taken together define a supersymmetric quantum mechanics, as originally defined by Witten. The supersymmetric quantum mechanics in imaginary time is a stochastic Brownian process with a drift which is an analytic function of the position. The conjugate potentials correspond to reversing the direction of the drift, and their properties are similar because they are related by a stochastic version of time-reversibility.
If W(x) goes to plus infinity at infinity (so that it actually defines a normalizable ground state), then the ground state is unique, and the conjugate potential has the exact same spectrum as the original potential, except it omits the lowest energy state. This, plus the form of the supercharge, gives exact solutions of many classes of quantum potentials.
Here are some simple W's which correspond to usual elementary quantum mechanics examples:

*

*W(x) = |x|^2 is the Hamonic oscillator in any dimension

*W(x) = |x| is the delta function potential in 1d, and the Coulomb potential in 3d

*W(x) = log(|cos(x)|) this gives the infinite hard wall

http://arxiv.org/abs/hep-th/9405029 has a bunch of more interesting examples. Any quantum mechanical potential which has closed form energy states is in this class.
A completely diffeent class of widely studied potentials are random potentials, as studied by Halperin and others, to understand Anderson localization.
Later Edit: The original paper by Anderson which started the random potential field is "Absence of diffusion in certain random lattices", and it's one of the great classics. The setup is a square lattice with a random potential at each site, an independent random number between -V and V. The continuum limit in one dimension, where the potential is a random gaussian at each point is analyzed by Halperin B. I. Halperin, Green ' s Functions for a Particle in a One-Dimensional Random Potential, Phys. Rev. 139 , A104 (1965). The field is enormous--- look up "localization" on google scholar. It includes "weak localization" effects, which were popular in the mid 90s because they imply that resistance can drop sharply in the presence of a magnetic field, because the perturbative precurser to the localization process is hindered.
A: The mostly used potentials are the parabolic potentials. You may have seen it in quantum harmonic oscillator with potential energy $\frac{1}{2}m\omega^2 x^2$.
The parabolic potential is of the form $\frac{1}{2}k x^2$ and corresponds to the force $F=-\frac{dV}{dx}=-kx$. It can be used as an approximation to potentials at local minima (stable equilibrium points), at which the second derivative is positive.
For example, suppose that you have a potential with local minimum at $x=0$ and $V''(0)>0$. Taking Taylor expansion of $V(x)$ near the $x=0$ you get: 
$V(x)=V(0)+V'(0)x+\frac{1}{2}V''(0)x^2+\cdot\cdot\cdot$ 
You can choose $V(0)=0$ and of course $V'(0)=0$ (local minimum at $x=0$). So finally you get $V(x)=\frac{1}{2}V''(0)x^2=\frac{1}{2}kx^2$, where $k=V''(0)>0$, as a good approximation of the $V(x)$ near the $x=0$.  
This parabolic approximation can be used for vibrations of diatomic or polyatomic molecules.
A: The Schrodinger Equation was heuristically derived to force classical trigonometric wave solutions in the space coordinates.  It was derived for constant potential function and assumed to also apply for variable potential functions.  Analytical solutions using separation of variables method limit potential functions to time independent potential functions.
There is an infinite manifold of theoretical time independent potential functions for which exact analytical solutions of the Schrodinger equation can be obtained by quadrature.  The challenge to theorists is to identify subset of these theoretical potential functions correspond to empirically observable processes.  At quantum scale space dimensions there may be forces that influence massive particle behavior/dynamics that have yet to be discovered. Gravity may play a significant role at quantum scale space dimensions that does not obey Newtonian inverse square law or General Relativity. Such force fields at quantum scale dimensions are inferable from observation of massive particle trajectories and the hidden potential functions found by simple differentiation. Entanglement of massive particles can be easily explained if quantum scale worm holes allow communication between particles that does not defy the limitation of the speed of light.
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