Planck Scale Physics of the Single Particle Schr\"{o}dinger Equation with Gravitational Self-Interaction

TL;DR

This paper explores a nonlinear modification of the Schrödinger equation incorporating gravitational self-interaction, revealing that particles near the Planck mass behave like black holes and violate superposition.

Contribution

It introduces a gravitationally self-interacting Schrödinger equation and analyzes its solutions, showing the transition to black hole behavior at the Planck mass.

Findings

Particles near Planck mass become black holes.

The equation admits self-bound stationary solutions.

Superposition is strongly violated for high-mass particles.

Abstract

We consider the modification of a single particle Schr\"{o}dinger equation by the inclusion of an additional gravitational self-potential term which follows from the prescription that the' mass-density'that enters this term is given by $m |\psi (\vec {r},t)|^2$, where $\psi (\vec {r}, t)$ is the wavefunction and $m$ is the mass of the particle. This leads to a nonlinear equation, the ' Newton Schrodinger' equation, which has been found to possess stationary self-bound solutions, whose energy can be determined exactly using an asymptotic method. We find that such a particle strongly violates superposition and becomes a black hole as its mass approaches the Planck mass.