A simple model to describe the low-temperature behaviour of some atoms and molecules: an application to the hydrogen atom

Abstract

A simple model to describe the low-temperature behaviour of some atoms and molecules is proposed and applied to the hydrogen atom. To this end, the low-temperature dynamics of the approach to equilibrium of the hydrogen atom is analysed by means of a standard Monte Carlo simulation. It is shown that, before approaching ionization, the atom may live for a long time in a quasi-equilibrium state whose duration increases exponentially for decreasing temperatures. Essentially, this effect is directly related to the low probability associated with the transition between the ground- and first-excited states, which demands an enormous amount of energy (75% of the whole energy spectrum). Therefore, for low temperatures, the atom may take a long time to overcome such an energy barrier. It is argued that the dynamical behaviour associated with the approach to equilibrium of some composite particles, characterized by an energy spectrum presenting an upper bound, preceded by the accumulation of an infinite number of levels—for which the hydrogen atom represents a prototype—can be described, at low temperatures, by a special class of q-oscillators. By suitably adjusting the deformation parameter q, characteristic of these q-oscillator systems, one obtains a dynamical behaviour at low temperatures which resembles that associated with the composite particle of interest. In order to reproduce the results of the hydrogen atom, the central idea is that this parameter may be set to a value, in such a way that the energy gap between the ground- and first-excited states coincide in the two systems. The method is illustrated by choosing q = 1/4, in which case one gets a remarkable agreement, from both qualitative and quantitative points of view, with the dynamical behaviour of the hydrogen atom. The conditions of applicability of the method are discussed.