Is Gravitational Mass of a Composite Quantum Body Equivalent to its Energy?

Abstract

We define gravitational mass operator of a hydrogen atom in the post-Newtonian approximation of the General Relativity and show that it does not commute with energy operator. Nevertheless, the equivalence between the expectation values of gravitational mass and energy is shown to survive for stationary quantum states. Inequivalence between gravitational mass and energy at a microscopic level reveals itself as unusual electromagnetic radiation, emitted by the atoms, supported and moved in the Earth gravitational field, which can be experimentally measured. Inequivalence between gravitational mass and energy at a macroscopic level results in time dependent oscillations of the expectation values of gravitational mass for mixed quantum states.