On the modification of Hamiltonians' spectrum in gravitational quantum mechanics

TL;DR

This paper investigates how a recently proposed Generalized Uncertainty Principle, consistent with various quantum gravity theories, modifies Hamiltonians and affects the energy spectra of quantum systems like particles in a box and harmonic oscillators.

Contribution

It introduces a GUP with both minimum length and maximum momentum, deriving new Hamiltonian terms and analyzing their effects on quantum systems' eigenenergies.

Findings

Corrections to eigenenergies are proportional to the square of the energy level for polynomial potentials.

The proportionality of energy corrections is exact for a particle in a box.

The study provides insights into quantum gravity effects on quantum mechanical spectra.

Abstract

Different candidates of Quantum Gravity such as String Theory, Doubly Special Relativity, Loop Quantum Gravity and black hole physics all predict the existence of a minimum observable length or a maximum observable momentum which modifies the Heisenberg uncertainty principle. This modified version is usually called the Generalized (Gravitational) Uncertainty Principle (GUP) and changes all Hamiltonians in quantum mechanics. In this Letter, we use a recently proposed GUP which is consistent with String Theory, Doubly Special Relativity and black hole physics and predicts both a minimum measurable length and a maximum measurable momentum. This form of GUP results in two additional terms in any quantum mechanical Hamiltonian, proportional to $\alpha p^3$ and $\alpha^2 p^4$, respectively, where $\alpha \sim 1/M_{Pl}c$ is the GUP parameter. By considering both terms as perturbations, we study two quantum mechanical systems in the framework of the proposed GUP: a particle in a box and a simple harmonic oscillator. We demonstrate that, for the general polynomial potentials, the corrections to the highly excited eigenenergies are proportional to their square values. We show that this result is exact for the case of a particle in a box.