# Hydrogen Atom and Time Variation of Fine-Structure Constant

## TL;DR

This paper investigates how the fine-structure constant, electron mass, and Planck constant vary over time within a de Sitter space-time framework, using the Dirac equation for hydrogen and an adiabatic approximation, aligning with current observations.

## Contribution

It introduces a novel approach to analyze time variation of fundamental constants in de Sitter space-time using the Dirac equation and adiabatic approximation, providing a cosmological context.

## Key findings

- Fine-structure constant varies with time.
- Electron mass and Planck constant also show time variation.
- Results align with current observational data.

## Abstract

In this paper, we have solved the de Sitter special relativistic ($\mathcal{SR}_{cR}$-) Dirac equation of hydrogen in the earth-QSO(quasar) framework reference by means of the adiabatic approach. The aspects of geometry effects of de Sitter space-time described by Beltrami metric are explored and taken into account. It is found that the $\mathcal{SR}_{cR}$-Dirac equation of hydrogen is a time dependent quantum Hamiltonian system. We provide an explicit calculation to justify the adiabatic approach in dealing with this time-dependent system. Since the radius of de Sitter sphere $R$ is cosmologically large, the evolution of the system is very slow so that the adiabatic approximation legitimately works with high accuracy. We conclude that the electromagnetic fine-structure constant, the electron mass and the Planck constant are time variations. This prediction of fine-structure constant is consistent with the presently available observation data. For confirming it further, experiments/observations are required.