Breakdown of the Equivalence between Energy Content and Weight in a Weak Gravitational Field for a Quantum Body

Abstract: It is shown that weight operator of a composite quantum body in a weak external gravitational field in the post-Newtonian approximation of the General Relativity does not commute with its energy operator, taken in the absence of the field. Nevertheless, the weak equivalence between the expectations values of weight and energy is shown to survive at a macroscopic level for stationary quantum states for the simplest composite quantum body - a hydrogen atom. Breakdown of the weak equivalence between weight and energy at a microscopic level for stationary quantum states can be experimentally detected by studying unusual electromagnetic radiation, emitted by the atoms, supported and moved in the Earth gravitational field with constant velocity, using spacecraft or satellite. For superpositions of stationary quantum states, a breakdown of the above mentioned equivalence at a macroscopic level leads to time dependent oscillations of the expectation values of weight, where the equivalence restores after averaging over time procedure.