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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'?

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You pretty much named them all (although one can of course study general properties of the equation for broader classes of potentials). So why are you asking if you know the answer? Also, what use is this information to you (because personally, I can't think of any use at all...)? –  Marek Jul 7 '11 at 8:45
    
I need to write a paper about Schrodinger equation, and discuss some examples of the potential energy(as many as possible). –  NGY Jul 7 '11 at 8:52
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You forgot periodical functions in your list. They are heavily used in solid state physics, where they correspond to the potential seen by an electron in a metallic crystal. –  Frédéric Grosshans Jul 7 '11 at 9:25
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There is also the delta-function potential. It might be used to model point impurities in a metal. Potentials involving the spin components can also be included, like the Zeeman term or spin-orbit interactions. There are many other possibilities. But in most modern research, one-particle potentials won't do the job. The stuff you mention is mainly on the undergrad QM course level and therefore very far away from most research involving QM. What do you exactly mean by "research"? –  Heidar Jul 7 '11 at 10:18
    
Thanks for your comments, Frédéric Grosshans and 4tnemele. My 'research' is same as what you mean(but as I know little about research in QM and other branches of physics, I could only mention those undergrad level). I just want to know potentials heavily used in research involving QM(or other branches of physics), and to summerize them. –  NGY Jul 7 '11 at 10:27
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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.

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Sometimes $V''(0)=0$ as well. Then harmonic approximation is no long valid. –  C.R. Jul 7 '11 at 11:15
    
"Sometimes $V''(0)=0$" Can you give an example? And why it's not valid? –  Andyk Jul 7 '11 at 14:29
    
A typical example is a quartic potential, $V(x) = a x^4.$ –  Gerben Jul 8 '11 at 8:36
    
You can take the Taylor expansion of $V(x)=a x^4$ at x=0, I don't see any problem with that. Of course it will be just an approximation of V(x) at particular point. –  Andyk Jul 8 '11 at 17:10
    
"You can take the Taylor expansion of $V(x)=ax^4$ at $x=0$" You can. But the $x^2$ term will have a co-efficient of zero, so the problem can not be approximated as harmonic. –  dmckee Jul 15 '11 at 14:50
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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 http://prola.aps.org/abstract/PR/v109/i5/p1492_1 "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.

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Hi Ron, can you give me some reference for random potentials? What theoretical results do we have for random potentials? –  felix Aug 15 '11 at 0:03
    
I added some entry points. The basic theoretical results are that all the energy eigenstates in one and two dimensions with an arbitrarily small random potential decay exponentially at long distances. In three dimensions and higher, there is a phase transition as you vary the strength of the randomness between localized and extended wavefunctions. There is very little known rigorously, as far as I know, but a huge amount known by physics standards of certainty. I added a few very old literature pointers in the answer itself. –  Ron Maimon Aug 15 '11 at 4:16
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