Generalized Lennard-Jones Potentials, SUSYQM and Differential Galois Theory

Abstract

In this paper, we start with proving that the Schrödinger 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 a differential equation. We study the 10−6 potential through DGT and SUSYQM, being one of the two partner potentials built with a superpotential of the form w(r)∝1/r⁵. 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 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 (2ν−2)−ν L-J potential. This analysis does not include the study of square integrable wave functions, excited states, and energies different than zero for the generalization of L-J potentials.