The effects of minimal length and maximal momentum on the transition rate of ultra cold neutrons in gravitational field

TL;DR

This paper investigates how a minimal length and maximal momentum, inspired by quantum gravity theories, modify the energy levels and transition rates of ultra cold neutrons in gravitational fields using a generalized uncertainty principle.

Contribution

It introduces a GUP-based Hamiltonian with additional momentum-dependent terms and analyzes their effects on quantum bouncer energy levels and transition rates.

Findings

Modified energy eigenvalues and eigenfunctions for the quantum bouncer.

Altered transition rates of ultra cold neutrons due to GUP effects.

Second-order GUP corrections considered in the analysis.

Abstract

The existence of a minimum observable length and/or a maximum observable momentum is in agreement with various candidates of quantum gravity such as string theory, loop quantum gravity, doubly special relativity and black hole physics. In this scenario, the Heisenberg uncertainty principle is changed to the so-called Generalized (Gravitational) Uncertainty Principle (GUP) which results in modification of all Hamiltonians in quantum mechanics. In this paper, following a recently proposed GUP which is consistent with quantum gravity theories, we study the quantum mechanical systems in the presence of both a minimum length and a maximum momentum. The generalized Hamiltonian contains two additional terms which are proportional to $\alpha p^3$ and $\alpha^2 p^4$ where $\alpha \sim 1/M_{Pl}c$ is the GUP parameter. For the case of a quantum bouncer, we solve the generalized Schrodinger equation in the momentum space and find the modified energy eigenvalues and eigenfunctions up to the second-order in GUP parameter. The effects of the GUP on the transition rate of ultra cold neutrons in gravitational spectrometers are discussed finally.