Year: 2007
Type: article
Abstract: We apply a singularity-resolution technique utilized in loop quantum gravity to the polymer representation of quantum mechanics on $\mathbb{R}$ with the singular $\ensuremath{-}1/|x|$ potential. On an equispaced lattice, the resulting eigenvalue problem is identical to a finite-difference approximation of the Schr\"odinger equation. We find numerically that the antisymmetric sector has an energy spectrum that converges to the usual Coulomb spectrum as the lattice spacing is reduced. For the symmetric sector, in contrast, the effect of the lattice spacing is similar to that of a continuum self-adjointness boundary condition at $x=0$, and its effect on the ground state is significant even if the spacing is much below the Bohr radius. Boundary conditions at the singularity thus have a significant effect on the polymer quantization spectrum even after the singularity has been regularized.
Authors: Viqar Husain, Jorma Louko, Oliver Winkler
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2037524850