Ground-state energy of a particle in a space with minimal length and minimal momentum

TL;DR

This paper establishes a lower bound on the ground-state energy of one-dimensional quantum systems in deformed space with minimal uncertainties, including harmonic and anharmonic oscillators.

Contribution

It derives a general expression for the minimal energy in deformed space and analyzes solution domains for various potentials.

Findings

Calculated ground-state energy for harmonic oscillator in deformed space.

Derived an equation for coordinate uncertainty at minimal energy.

Identified parameter domains where solutions exist for anharmonic potentials.

Abstract

In this article, we derived a rigorous lower bound on the ground-state energy for a class of one-dimensional quantum systems in deformed space with minimal coordinate and momentum uncertainties, representing the absolute minimum energy that is physically attainable. We considered a harmonic oscillator in such a space and calculate its ground-state energy. We generalized the problem to an a sufficiently broad class of potentials, deriving an equation for the coordinate uncertainty corresponding to the minimal energy, which can be solved numerically. Using a linear approximation in the deformation parameters, we obtained a general expression for the ground-state energy. We determined the domain of existence of solutions for the anharmonic oscillator potential with respect to the deformation parameters.