Schr\"odinger-Newton "collapse" of the wave function

TL;DR

This paper investigates the Schr"odinger-Newton equation's role in modeling quantum-gravity coupling, showing that wave packets can partially collapse or spread depending on mass, indicating potential deviations from standard quantum mechanics at certain scales.

Contribution

It provides numerical analysis of the Schr"odinger-Newton equation, clarifying conditions under which wave function collapse or spreading occurs, and discusses implications for quantum mechanics and gravity.

Findings

Wave packets collapse to a ground state for masses above a threshold.

Wave packets spread like free particles for smaller masses.

Significant deviations from linear quantum mechanics are predicted at certain scales.

Abstract

It has been suggested that the nonlinear Schr\"odinger-Newton equation might approximate the coupling of quantum mechanics with gravitation, particularly in the context of the M{\o}ller-Rosenfeld semiclassical theory. Numerical results for the spherically symmetric, time-dependent, single-particle case are presented, clarifying and extending previous work on the subject. It is found that, for a particle mass greater than $\sim 1.14(\hbar^2/(G\sigma))^{1/3}$, a wave packet of width $\sigma$ partially "collapses" to a groundstate solution found by Moroz, Penrose, and Tod, with excess probability dispersing away. However, for a mass less than $\sim 1.14(\hbar^2/(G\sigma))^{1/3}$, the entire wave packet appears to spread like a free particle, albeit more slowly. It is argued that, on some scales (lower than the Planck scale), this theory predicts significant deviation from conventional (linear) quantum mechanics. However, owing to the difficulty of controlling quantum coherence on the one hand, and the weakness of gravity on the other, definitive experimental falsification poses a technologically formidable challenge.