Planck scale physics of the single-particle Schrödinger equation with gravitational self-interaction
VIKRAM SONI
National Physical Laboratory, K.S. Krishnan Marg, New Delhi 110 016, India
Abstract. We consider the modification of a single-particle Schrödinger 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 |f (\vec{r},t)|2, where (f \vec{r},t) is the wave function and m is the mass of the particle. This leads to a nonlinear equation, the `Newton--Schrödinger' equation, which has been found to possess stationary self-bound solutions, whose energy can be determined using an asymptotic method. We find that such a particle strongly violates the superposition principle and becomes a black hole as its mass approaches the Planck mass.
Keywords. Quantum mechanics; gravitation; state reduction.
PACS Nos 03.65.Ta; 04.20; 04.70; 04.90