Minimal Length and the Quantum Bouncer: A Nonperturbative Study

TL;DR

This paper investigates the energy spectrum of a quantum bouncer under the Generalized Uncertainty Principle using nonperturbative methods, providing a consistent semiclassical and quantum mechanical analysis.

Contribution

It introduces a new self-adjoint representation of the GUP-deformed commutation relation that yields consistent energy spectra with existing quantum mechanical results.

Findings

Both representations produce identical semiclassical energy spectra.

The modified spectra align well with quantum mechanical calculations.

The study offers a nonperturbative approach to GUP effects in quantum systems.

Abstract

We present the energy eigenvalues of a quantum bouncer in the framework of the Generalized (Gravitational) Uncertainty Principle (GUP) via quantum mechanical and semiclassical schemes. In this paper, we use two equivalent nonperturbative representations of a deformed commutation relation in the form [X,P]=i\hbar(1+\beta P^2) where \beta is the GUP parameter. The new representation is formally self-adjoint and preserves the ordinary nature of the position operator. We show that both representations result in the same modified semiclassical energy spectrum and agrees well with the quantum mechanical description.