One Electron Atom in Special Relativity with de Sitter Space-Time Symmetry

The de Sitter invariant Special Relativity (dS-SR) is a SR with constant curvature, and a natural extension of usual Einstein SR (E-SR). In this paper, we solved the dS-SR Dirac equation of Hydrogen by means of the adiabatic approach and the quasi-stationary perturbation calculations of QM. Hydrogen atoms are located on the light cone of the Universe. FRW metric and $\Lambda$CDM cosmological model are used to discuss this issue. To the atom, effects of de Sitter space-time geometry described by Beltrami metric are taken into account. The dS-SR Dirac equation turns out to be a time dependent quantum Hamiltonian system. We revealed that: 1,The fundamental physics constants $m_e,\;\hbar,\;e$ variate adiabatically along with cosmologic time in dS-SR QM framework. But the fine-structure constant $\alpha\equiv e^2/(\hbar c)$ keeps to be invariant; 2,$(2s^{1/2}-2p^{1/2})$-splitting due to dS-SR QM effects: By means of perturbation theory, that splitting $\Delta E(z)$ were calculated analytically, which belongs to $\mathcal{O}(1/R^2)$-physics of dS-SR QM. Numerically, we found that when $|R|\simeq \{10^3 Gly,\;10^4 Gly,\;10^5 Gly\;\}$, and $z\simeq \{1,\;{\rm or}\;2\}$, the $\Delta E(z)>> 1{\rm (Lamb\; shift)}$. This indicate that for these cases the hyperfine structure effects due to QED could be ignored, and the dS-SR fine structure effects are dominant. This effect could be used to determine the universal constant $R$ in dS-SR, and be thought as a new physics beyond E-SR.