Planck Scale Physics of the Single Particle Schrödinger Equation with Gravitational Self-Interaction

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 $m |ψ(\vec {r},t)|^2$, where $ψ(\vec {r}, t)$ 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.

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