Space-Curvature Effects in Atomic and Molecular-Structure Calculations

Abstract

The consideration of the curvature of the physical space in the formulation of Quantum Physics is an old problem which has been studied for a long time [1–4] and, in the field of Atomic and Molecular Physics, has been reinvestigated or reviewed, more recently, by several authors [5–8]. In principle, owing to the fundamental link between physics and geometry, it should not be very surprising that even the slightest modifications of the structure of physical space could modify spectra significantly and specifically. This is remarkably well illustrated in a pioneering work of Schrödinger [9] who solved exactly, in the spherical or Einstein universe, the non relativistic one-electron wave equation bearing his name and placed on record how, when the usual Euclidean flat-space is substituted by a spherical three-space, the continuous hydrogenic spectrum is resolved into an intensely crowded line spectrum. Since the mathematical nature of the hydrogenic wavefunction is no more intricate in spherical three-space than in flat space [10, 11], it is possible to build up a tractable “curved orbital” model (non relativistic or relativistic) capable of exploring, at least roughly, some space-curvature modifications of atomic and molecular spectra. Working in that geometrically simple space, i.e. a three-dimensional hypersphere of radius R embedded in a Euclidean four-space, allows us to keep a direct parallelism between the “curved” (1/R ≠ 0) and the “flat” (1/R = 0) results and an easy extension to the many electron case.