Square-well potential, finite, particle

Imagine that a particle of mass m is free to move along the x axis between x = 0 and x = L, with no change in potential (set F = 0 for 0 < x < L). At x = 0 and L and at all points heyond these limits the particle encounters an infinitely repulsive barrier (V = oo for X <0, X > L). The situation is illustrated in Fig. 2-1. Because of the shape of this potential, this problem is often referred to as a particle in a square well or a particle in a box problem. It is well to bear in mind, however, that the situation is really like that of a particle confined to movement along a finite length of wire. 

Fig. 2.2. Average electrostatic potential mc at the position of the methane-like Lennard-Jones particle Me as a function of its charge q. mc contains corrections for the finite system size. Results are shown from Monte Carlo simulations using Ewald summation with N = 256 (plus) and N = 128 (cross) as well as GRF calculations with N = 256 water molecules (square). Statistical errors are smaller than the size of the symbols. Also included are linear tits to the data with q < 0 and q > 0 (solid lines). The fit to the tanh-weighted model of two Gaussian distributions is shown with a dashed line. Reproduced with permission of the American Chemical Society...

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