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 ‘massdensity’ that enters this term is given by m|ψ($$\overrightarrow r $$,t)|2, where ψ($$\overrightarrow r $$,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.
Volume 96, 2022
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