Quantum Potentials with q-Gaussian Ground States

TL;DR

This paper identifies families of spherically symmetric quantum potentials in D dimensions that have ground states with q-Gaussian wavefunctions, linking them to maximum entropy principles and Coulomb-like behavior.

Contribution

It introduces new quantum potentials with q-Gaussian ground states, extending the understanding of their form and asymptotic behavior in relation to Coulomb potentials.

Findings

Potentials with q-Gaussian ground states in momentum space include Coulomb potential as a special case.

All such potentials behave like Coulomb potential asymptotically for large r when 0<q<1.

Ground states can be described using maximum entropy S_q power-law entropies.

Abstract

We determine families of spherically symmetrical $D$-dimensional quantum potential functions $V(r)$ having ground state wavefunctions that exhibit, either in configuration or in momentum space, the form of an isotropic $q$-Gaussian. These wavefunctions admit a maximum entropy description in terms of $S_q$ power-law entropies. We show that the potentials with a ground state of the $q$-Gaussian form in momentum space admit the Coulomb potential $-1/r$ as a particular instance. Furthermore, all these potentials behave asymptotically as the Coulomb potential for large $r$for all values of the parameter $q$ such that $0<q<1.$