Generalized Lennard-Jones Potentials, SUSYQM and Differential Galois Theory

TL;DR

This paper investigates the integrability of Schrödinger equations with Lennard-Jones potentials using differential Galois theory and SUSYQM, finding nonintegrability for the classical potential but integrability at zero energy for a modified potential, and proposing an alternative for modeling.

Contribution

It demonstrates the nonintegrability of the classical 12-6 Lennard-Jones potential and introduces a 10-6 potential as an integrable alternative, analyzing their physical implications.

Findings

12-6 potential is nonintegrable in DGT for all energies.

10-6 potential is integrable at zero energy.

Good agreement of second virial coefficient at low temperatures.

Abstract

In this paper we start with proving that the Schr\"odinger equation (SE) with the classical $12-6$ Lennard-Jones (L-J) potential is nonintegrable in the sense of the differential Galois theory (DGT), for any value of energy; i.e., there are no solutions in closed form for such differential equation. We study the $10-6$ potential through DGT and SUSYQM; being it one of the two partner potentials built with a superpotential of the form $w(r)\propto 1/r^5$. We also find that it is integrable in the sense of DGT for zero energy. A first analysis of the applicability and physical consequences of the model is carried out in terms of the so called De Boer principle of corresponding states. A comparison of the second virial coefficient $B(T)$ for both potentials shows a good agreement for low temperatures. As a consequence of these results we propose the $10-6$ potential as an integrable alternative to be applied in further studies instead of the original $12-6$ L-J potential. Finally we study through DGT and SUSYQM the integrability of the SE with a generalized $(2\nu-2)-\nu$ L-J potential. This analysis do not include the study of square integrable wave functions, excited states and energies different than zero for the generalization of L-J potentials.