Cloud of virtual photons in the ground state of the hydrogen atom.

Abstract

A spinless, nonrelativistic hydrogen atom coupled to an electromagnetic field is considered. The interaction is taken in the minimal-coupling form, and the ground state of the coupled system is obtained by straightforward perturbation theory. The form of the cloud of virtual photons surrounding the atom is studied through the quantum-mechanical average on this state of an appropriately defined coarse-grained energy-density (CGED) operator W(r\ensuremath{\rightarrow}). The properties of W(r\ensuremath{\rightarrow}) are studied in order to show that this operator can give a reliable description of the shape of the virtual photon cloud. The quantum-mechanical average of W(r\ensuremath{\rightarrow}) is obtained analytically and exactly, and the CGED is shown to consist of an infinite number of spherically symmetric contributions, each originating from the set of virtual transitions induced by the vacuum fluctuations between the bare atomic ground state and all the bare atomic eigenstates belonging to a given subshell. This yields a shell and subshell structure for the virtual photon cloud, each of them characterized by a different behavior of the CGED as a function of the distance from the atom. The details of this structure are studied both analytically and numerically, and the results obtained are compared to those pertaining to virtual clouds in other fields of physics.