Nonlinear Schr\"odinger equation from generalized exact uncertainty principle

TL;DR

This paper derives a nonlinear Schr"odinger equation inspired by the generalized uncertainty principle, incorporating gravitational effects into quantum mechanics, and explores its implications for minimal length scales and quantum gravity phenomenology.

Contribution

It extends the exact uncertainty principle approach to include gravitational effects, resulting in a novel nonlinear Schr"odinger equation with unique properties.

Findings

Solutions respect gravitational minimal length scale

Equation maintains separation of non-interacting systems

No modification of free-particle dispersion relation predicted

Abstract

Inspired by the generalized uncertainty principle (GUP), which adds gravitational effects to the standard description of quantum uncertainty, we extend the exact uncertainty principle (EUP) approach by Hall and Reginatto [J. Phys. A: Math. Gen. (2002) 35 3289], and obtain a (quasi)nonlinear Schr\"odinger equation. This quantum evolution equation of unusual form, enjoys several desired properties like separation of non-interacting subsystems or planewave solutions for free particles. Starting with the harmonic oscillator example, we show that every solution of this equation respects the gravitationally-induced minimal position uncertainty proportional to the Planck length. Quite surprisingly, our result successfully merges the core of classical physics with non-relativistic quantum mechanics in its extremal form. We predict that the commonly accepted phenomenon, namely a modification of a free-particle dispersion relation due to quantum gravity might not occur in reality.