An algebraic approach to gravitational quantum mechanics

TL;DR

This work develops an algebraic, coordinate-based approach to gravitational quantum mechanics built on a deformed Heisenberg algebra that imposes a minimal length $\lambda$. By introducing a GUP derivative $D_x$ and exploring multiple representations, the authors derive exact and approximate tools (including the GUP exponential and a GUP Fourier transform) to analyze Gaussian wave packets, their time evolution, and bound-state spectra in simple potentials. The study yields concrete results: a finite number of bound states in a box with size $L$ (with no states when $L<2\lambda$) and a finite, parameter-dependent set of bound states for a finite-depth well, including special closed-form energies for particular choices of $a$ and $V_0$. Overall, the approach demonstrates how minimal-length physics, encoded via the GUP, alters dispersion, uncertainty relations, and spectral properties in canonical quantum-mechanical settings.

Abstract

Most approaches towards a quantum theory of gravitation indicate the existence of a minimal length scale of the order of the Planck length. Quantum mechanical models incorporating such an intrinsic length scale call for a deformation of Heisenberg's algebra resulting in a generalized uncertainty principle and constitute what is called gravitational quantum mechanics. Utilizing the position representation of this deformed algebra, we study various models of gravitational quantum mechanics. The free time evolution of a Gaussian wave packet is investigated as well as the spectral properties of a particle bound by an external attractive potential. Here the cases of a box with infinite walls and an attractive potential well of finite depth are considered.

Figures (2)

• Figure 1: The graphical presentation of the even solutions of \ref{['cond1']} for fixed $\gamma =10$ and $a/\lambda$ varying from $1,5,10,20$ from left upper corner till right lower one. The intersections of the black line with blue lines indicate the even solutions to the eigenvalue problem. The red lines indicate the undeformed case with $\lambda = 0$.
• Figure 2: Same as figure \ref{['Fig1']} but for the odd solutions of \ref{['cond1']} with same set of parameters. The case $a=\lambda$ in the left upper graph does not exhibit an odd eigenstate as discussed in the text.