Particle in a cavity in one-dimensional bandlimited quantum mechanics

TL;DR

This paper investigates how the generalized uncertainty principle (GUP) affects low-energy states of a particle in a cavity within one-dimensional bandlimited quantum mechanics, revealing leading order effects of GUP on energy levels.

Contribution

It introduces a perturbation method to analyze GUP effects in bandlimited quantum mechanics, accounting for both Hamiltonian modifications and UV mode restrictions.

Findings

Leading GUP effects are of first order from UV mode restrictions.

Second order effects arise from Hamiltonian modifications.

Energy shifts are proportional to the ratio of Planck length to cavity size.

Abstract

The effects of the generalized uncertainty principle (GUP) on the low-energy stationary states of a particle moving in a cavity with no sharp boundaries are determined by means of the perturbation expansion in the framework of one-dimensional bandlimited quantum mechanics. A realization of GUP resulting in the existence of a finite ultraviolet (UV) wave-vector cutoff $K\sim 1/\ell_P$ (with the Planck length $\ell_P$) is considered. The cavity of the size $\ell \gg \ell_P$ is represented by an infinitely deep trapezoid-well potential with boundaries smeared out in a range $R$ satisfying the inequalities $\ell\gg R\gtrsim \ell_P$. In order to determine the energy shifts of the low-lying stationary states, the usual perturbation expansion is reformulated in a manner that enables one to treat consistently order-by-order the direct and indirect GUP effects, i.e., those due to the modification of the Hamiltonian and the lack of the UV modes, respectively. It is shown that the leading terms of the indirect and the direct GUP effects are of the first and second order, respectively, in the small parameter $\ell_P/\ell$ in agreement with our previous finding in a more naive approach [1].