Equivalence between Gravitational Mass and Energy for a Quantum Body at a Macroscopic Level

TL;DR

This paper demonstrates that the equivalence of gravitational mass and energy holds for stationary quantum states of a hydrogen atom, even with relativistic corrections, despite the non-commuting nature of the operators.

Contribution

It introduces a quantum operator for gravitational mass and proves its expectation value equals energy for stationary states, including relativistic effects.

Findings

Expectation value of gravitational mass equals energy in stationary states.

Relativistic corrections do not break the mass-energy equivalence.

Gravitational mass operator does not commute with energy operator.

Abstract

We define passive gravitational mass operator of the simplest composite quantum body - a hydrogen atom - to be proportional to its weight operator in a weak gravitational field. Although it does not commute with energy operator, taken in the absence of the field, the equivalence between the expectation value of passive gravitational mass and energy is shown to survive for stationary quantum states. All the so-called relativistic corrections to electron energy in a hydrogen atom are taken into account in the calculations.