Nonlinear Schrödinger equation from generalized exact uncertainty principle

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 Schrödinger 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.

References in corpus (4)

• Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrodinger equation in the radial case
• Gravitational Effects in Quantum Mechanics
• Black Hole Corrections due to Minimal Length and Modified Dispersion Relation
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Cited by in corpus (3)

• Detecting a logarithmic nonlinearity in the Schrödinger equation using Bose-Einstein condensates
• On entropic uncertainty relations in the presence of a minimal length
• A framework for nonrelativistic isotropic models based on generalized uncertainty principles