Is an Electron a Charge Cloud? A Reexamination of Schrödinger’s Charge Density Hypothesis

Abstract

This article re-examines Schrödinger's charge density hypothesis, according to which the charge of an electron is distributed in the whole space, and the charge density in each position is proportional to the modulus squared of the wave function of the electron there. It is demonstrated that the charge distribution of a quantum system can be measured by protective measurements as expectation values of certain observables, and the results as predicted by quantum mechanics confirm Schrödinger's original hypothesis. Moreover, the physical origin of the charge distribution is also investigated. It is argued that the charge distribution of a quantum system is effective, that is, it is formed by the ergodic motion of a localized particle with the charge of the system.

Electron Charge Density: A Clue from Quantum Chemistry for Quantum Foundations

“…6 When presenting this method, you commonly see authors employing Schrödinger's role for amplitude-squared and speaking of electrons as clouds of charge. 7 In this section, we will see why that picture is appealing as a way of understanding the ground state energy in the Hartree-Fock Method.…”

“…But, Bohmian mechanics does not vindicate the picture that Levine and Pauling paint (see section 4.3). Still, one may seek an alternative dynamics for which −e times the amplitude-squared of the electron wave function gives the time-averaged charge density (an approach recently advocated by Gao [5][6][7][8]).…”

“…What kind of density is the electron density? When presenting density functional theory, some authors describe the electron density as an expected number density (7) and others as a charge density (6). Here, we will treat it as a density of charge.…”

“…30 Although Levine [27, sec. 16.5] calls the electron density an "electron probability density," he does not mean by this that integrating the electron density (7) over any spatial region gives the probability of finding at least one electron in that region. For sufficiently small regions, the probability of finding more than one electron in the region is negligible and the integrated electron density over that region influential textbook on density functional theory, initially describe the electron density as a number density, giving no indication that it is only an expected number density and thus appearing to interpret it as an actual number density.…”

“…Physicists and philosophers have put forward a number of interpretations of quantum mechanics that seek to be clear about the ontology of the theory (what exists according to the theory), among which we will focus here on Ghirardi-Rimini-Weber (GRW) theory, the many-worlds interpretation, Bohmian mechanics, and the recent proposals by Hall et al [3], Sebens [4], and Gao [5][6][7][8]. Such attempts to clarify the ontology of quantum mechanics do not always include a charge density like the one Schrödinger proposed, though we will see in section 4 that it can often be added.…”