About the proof that the s-wave functions of the hydrogen atom obey the Ehrenfest equation of motion

Abstract

Applying the concept of quantum hydrodynamics on a hydrogen atom, one can derive an equation of motion which describes the time derivative of the mass current density for the relative motion of the electron and the proton. Since this derivation can be performed with the Ehrenfest theorem, we call this equation the Ehrenfest equation of motion. As quantum hydrodynamics is consistent with Schrödinger’s quantum mechanics, the stationary s-wave functions satisfy both the Schrödinger equation and the Ehrenfest equation of motion. It is the purpose of this paper to prove that the stationary s-wave functions satisfy the Ehrenfest equation of motion. From the quantum hydrodynamical point of view, this result for s-wave functions can be interpreted that the Coulomb force density that attracts the electron to the proton is compensated by a quantum force density that is related to the dispersion of the probability density.