Disfavoring the Schrödinger-Newton equation in explaining the emergence of classicality

The main goal of this paper is to provide some insight into how promising the Schrödinger-Newton equation would be to explain the emergence of classicality. Based on the similarity of the Newton and Coulomb potentials, we add an electric self-interacting term to the Schrödinger-Newton equation for the hydrogen atom. Our results rule out the possibility that single electrons self-interact through their electromagnetic field. Next, we use the hydrogen atom to get insight into the intrinsic difficulty of testing the Schrödinger-Newton equation itself and conclude that the Planck scale must be approached before sound constraints are established. Although our results cannot be used to rule out the Schrödinger-Newton equation at all, they might be seen as disfavoring it if we base our reasoning on the resemblance between the gravitational and electromagnetic interactions at low energies.

Figure 1

Plot of the radial probability density $F\left(r\right)$ for the ground state obtained from Eq. (4) (solid line) and Eq. (10) (dashed line).

Figure 2

Plot of the radial probability density $F\left(r\right)$ for the first excited state obtained from Eq. (4) (solid line) and Eq. (10) (dashed line).

Figure 3

Plot of the radial probability density $F\left(r\right)$ for the second excited state obtained from Eq. (4) (solid line) and Eq. (10) (dashed line).