Quantum mechanics of electromagnetically bounded spin-1/2 particles in an expanding universe: I. Influence of the expansion

Abstract

The quantum mechanically described electron in an external electromagnetic field, both embedded in an expanding universe with shear, is discussed. This is important for the fundamental question if a quantum mechanically treated atomic clock in curved space-time (based on a hydrogen atom) shows proper or gravitational time. Furthermore, contradictory results reported by other authors seem to imply that quantum mechanics cannot be reconciled with curved space-time. It is shown that this is not the case for expanding Robertson-Walker universes. As basis, in this paper a Hilbert space formulation of the problem with special regard to the Hamiltonian is given. The respective influence of the cosmic expansion and the intrinsic and extrinsic curvatures of the cosmic hypersurfaces on bound quantum mechanical systems is treated in general. For the special case of an expanding 3-flat (ε=0) Robertson-Walker universe it is shown that the energy levels of a hydrogen atom agree completely with the one in 4-flat space-time, so that in this case the hydrogen atom can be taken as atomic clock showing proper time.