Starting from the coupling of a relativistic quantum particle to the curved Schwarzschild space time, we show that the Dirac-Schwarzschild problem has bound states and calculate their energies including relativistic corrections. Relativistic effects are shown to be suppressed by the gravitational fine-structure constant αG=Gm1m2/(ℏc), where G is Newton's gravitational constant, c is the speed of light, and m1 and m2 ≫ m1 are the masses of the two particles. The kinetic corrections due to space-time curvature are shown to lift the familiar (n,j) degeneracy of the energy levels of the hydrogen atom. We supplement the discussion by a consideration of an attractive scalar potential, which, in the fully relativistic Dirac formalism, modifies the mass of the particle according to the replacement m→m(1-λ/r), where r is the radial coordinate. We conclude with a few comments regarding the (n,j) degeneracy of the energy levels, where n is the principal quantum number, and j is the total angular momentum, and illustrate the calculations by way of a numerical example.
U. D. Jentschura, "Fine-structure constant for gravitational and scalar interactions," Physical Review A - Atomic, Molecular, and Optical Physics, vol. 90, no. 2, pp. 022112-1-022112-7, American Physical Society (APS), Aug 2014.
The definitive version is available at https://doi.org/10.1103/PhysRevA.90.022112
Keywords and Phrases
Atoms; Electron Energy Levels; Quantum Theory; Fine Structure Constants; Gravitational Constant; Principal Quantum Numbers; Quantum Particles; Radial Coordinates; Relativistic Correction; Relativistic Effects; Scalar Potential; Gravitation
International Standard Serial Number (ISSN)
Article - Journal
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