Abstract
Previous results concerning the Dirac energies and wave functions in a space of constant positive curvature are used to derive their counterparts in a space of constant negative curvature. In particular, it is shown that, when switching from a spherical closed space to a hyperbolic open space, the sign of the curvature-induced splittings and shifts in the hydrogenic fine-structure spectrum is reversed.
Highlights
In order to complement our previous investigation on space curvature effects in atomic structure calculations,[1-3] it should be instructive to comparatively examine the modifi cations of the hydrogenic Dirac energies and wave functions when the physical space is assumed to be a hyperbolic three-space with constant negative curvature (Milne's open universe) instead of a spherical three-space (Einstein's closed universe with constant positive curvature 1/R)
As pointed out in the preceding section, the expressions of the fine-structure energies in the open space of constant curvature can be directly derived from their analogous expressions in the closed space by merely transposing R into iR
We have summarized some results concerning atomic structure calculations in a space of constant negae--e-(Za2/R) tive curvature
Summary
In order to complement our previous investigation on space curvature effects in atomic structure calculations,[1-3] it should be instructive to comparatively examine the modifi cations of the hydrogenic Dirac energies and wave functions when the physical space is assumed to be a hyperbolic three-space with constant negative curvature (Milne's open universe) instead of a spherical three-space (Einstein's closed universe with constant positive curvature 1/R). In the nonrelativis tic scheme,[1] we know that the final expressions of the energies, in each case, can be obtained one from each other by the formal change R - iR. This can be inferred from inspection of the wave equation. The modifications of the expressions of the hydrogenic fine-structure energies as well as the ex pressions of the wave functions (relativistic and nonrela tivistic) are given and discussed
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