# About constructing potential energy functions

There are many classical systems with different potential functions. My problem is that I do not understand how one can construct a certain potential function for a certain system. Are there any references I can look up in order to understand how the potential functions must look for a system that I am interested in building?

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Are you interested in a mechanical system that simulates the effect of an arbitrary potential or do you want a free space particle to experience a free space potential? Do you want the particle to travel in 1, 2 or 3 dimensions? –  FrankH Jan 29 '12 at 0:01

If it's possible to construct a classical potential $V(\vec x)$ for a system, then certainly the gradient of the potential should conincide with the force at every point. Likewise you can think of its values as integrating up the work $\int F\ \text{d}\vec s$ from the lowest point $\vec{x}_0$ of potential energy $V(\vec x_0)\equiv V_0$. And if there is an equilibrium situation, then the potential will have a local mimium there. If you make a second order expansion in that well, then you'll always end up with a harmonic oscillator.

A major guideline is also that the potential will have the same symmetries as the system. If you sit in the center and the whole problem is rotationally symmetric, then a potential of the form $V(|\vec x|)$ will not come as a surprise.

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It depends on the situation. Firstly, if you the look at the points of equilibrium, you know the maxima and minima depending on the nature of the equilibrium (it may be either stable, unstable or metastable equilibrium). As Nick has pointed out in the earlier answer, it depends on the symmetries of the system. Forces which are rotation invariant are generally functions of $|\mathbf{x}|$. (Gravitational, Electromagnetic potential are of the form $1/r$) Sometimes, they do have quantum origin too. Say, the force between two neutral atoms or molecules also called the Van der Waals force goes as $1/r^{6}$. The force is generally modeled by the Lennard-Jones potential which looks as follows:

The Lennard-Jones potential is actually \begin{equation*} V_{LJ}(r) = \frac{A}{r^{12}}-\frac{B}{r^6} \end{equation*} To get the attractive $1/r^{6}$ term, you need perturbation theory and the assumption that all inter-molecular interactions are electromagnetic in origin. The details of the calculation is given in Sakurai, Chapter 5 (Approximation Methods). The repulsive $1/r^{12}$ term comes from Pauli repulsion. As it seems, the potential is minimum at the van der Waals radius which gives the most stable configuration. This models the real interaction very well and is also used for the nice functional form it has. If I remember correctly, this was also used in the study of the Drude Model (Ashcroft and Mermin) and it gives right answers. So, it is not always right to assume that a classical looking potential will have a classical origin. Hence, a lot of factors goes in while construction of the potential: Equilibrium points, symmetries of the problem, constraints (if any, at all), quantum mechanics, experimental data etc. The skill lies how nicely you can approximate it with a well-behaving function and predict results (as I pointed out earlier, the Drude Model is partially successful in doing it)

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