Quantum bound states with zero binding energy - Comment

Sergio A. Hojman, Darío Núñez

A. Rokhlenko

John Lekner

The behaviour of two bodies that are just bound or nearly bound is discussed. A class of potentials is given for which Schrödinger's equation has exact solutions at critical binding (zero binding energy). This class includes the known solution for the 6–10 potential. For a general potential characterized by a coupling parameter α, it is shown that the bound state energy tends to zero as -(α - α0)2, where α0 is the critical value of the coupling parameter. Small energy scattering of atoms which are near critical binding (e.g. helium atoms) is examined. It is shown that determination of the total cross-section up to terms of order k 2 is in principle sufficient to distinguish between bound and virtual states of the diatomic molecule.

Royce K. P. Zia, Reinhard Lipowsky, D. M. Kroll

Potentials with repulsive power behavior at large distances may or may not have a zero-energy bound state, depending only on the power p and, if p=2, the strength. For the case of no zero-energy bound state, we further study how the state unbinds as the attractive, small r part of the potential is tuned to let the binding energy go to zero. Although motivated by the physics of wetting, this problem is well suited for senior or graduate level quantum mechanics courses by giving a novel perspective of solutions to one-dimensional Schroedinger equations.

Ruth E. Kastner

Richard Froese