Hydrogen energy-level shifts induced by the atom motion: Crossover from the Lamb shifts to the motion-induced shifts

Abstract

When the hydrogen atom moves, the proton current generates a magnetic field which interacts with the hydrogen electron. A simple analyze shows that this interaction between the hydrogen momentum and the electron is of order of α 3 v c m, where α is the fine structure constant, v is the atom velocity, c is the speed of light, and m is the electron mass. Using the Bethe-Salpeter equation, the two velocity-dependent operators of this order are derived for the hydrogen velocity v c << α. As well known, the degeneracy of the energy levels with the same principal quantum number, n, and the same quantum number of the total angular momentum, j, but the different orbital angular momenta l = j ± 1/2 is removed by the radiative corrections (the Lamb shift) that are proportional to α 5 m. It is shown that the velocity-dependent perturbation interactions remove this degeneracy as well. There is, however, an important difference between the Lamb shifts and the energy-levels shifts induced by the atom motion. The Lamb shift is the diagonal correction to the energy separately for each of the degenerate states. The the velocity-dependent perturbation interactions result in the off-diagonal energy corrections between the mutually degenerate states. The joint effect of these two perturbations which are essentially different in their origin, is analyzed. Given their order of magnitude, the crossover from the Lamb shifts to the motion-induced shifts should occur at the atom velocity v = kα 2 c, where k >> 1 is a numerical factor depended on n and j. An experiment used the orbital motion of the Earth, is proposed to test the theory developed.Keywords: hydrogen-like atom; the energy-level degeneracy; the Bethe-Salpeter equation; the Lamb shift; atom-motion-induced shifts PACS numbers:03.65. Pm, 31.30.jf Here the α e matrix is used in the standard representation, the subscript e means the action of the Pauli matrices on the electron spin, and V (r) = − e 2 r is the Coulomb interaction between the electron and proton.

History and Some Aspects of the Lamb Shift

“…New developments in calculations include simplifications to the Bethe-Salpeter equation for a system with masses that are very different, like the proton and electron [57,65,[87][88][89][90]. The simplifications are described as effective potential methods, and the "on the mass shell" approach [5].…”

The Invariant-Mass-Based Equation for Bound States of the Hydrogen and the Helium Atoms

“…In the frame of reference associated with a fixed proton (r p = 0), the bispinor wave function of the electron depends only on its radius-vector, ψ(r e ). As a result, the two-particle systems is described by the single-particle bound states for the Dirac equation with the external field [5][6][7][8][9]. For systems of many fermions, the motion of each fermion is described by the Dirac equation in an external field [10].…”