Hydrogen atom on curved noncommutative space

TL;DR

This paper calculates the hydrogen atom spectrum on a curved noncommutative space, revealing how noncommutativity introduces corrections to the fine structure while preserving rotational symmetry and degeneracy.

Contribution

It introduces a model of hydrogen on curved noncommutative space with a specific antisymmetric field, showing how noncommutativity affects the energy spectrum and nonlocality.

Findings

Noncommutativity causes corrections to the hydrogen fine structure.

Rotational symmetry and degeneracy are preserved in this model.

Nonlocality is quantified by a specific uncertainty relation.

Abstract

We have calculated the hydrogen atom spectrum on curved noncommutative space defined by the commutation relations $\left[ \hat {x}^{i},\hat{x}^{j}\right] =i\theta\hat{\omega}^{ij}\left( \hat {x}\right) $, where $\theta$ is the parameter of noncommutativity. The external antisymmetric field which determines the noncommutativity is chosen as $\omega^{ij}(x) =\varepsilon^{ijk}{x}_{k}f\left( {x_i}x^{i}\right) $. In this case the rotational symmetry of the system is conserved, preserving the degeneracy of the energy spectrum. The contribution of the noncommutativity appears as a correction to the fine structure. The corresponding nonlocality is calculated: $\Delta x\Delta y \geq \frac{\theta^2}{4} |m\langle f^2\rangle| $, where $m$ is a magnetic quantum number.