We derive the effect of the Schrodinger--Newton equation, which can be considered as a non-relativistic limit of classical gravity, for a composite quantum system in the regime of high energies. Such meson-antimeson systems exhibit very unique proper
ties, e.g. distinct masses due to strong and electroweak interactions. We find conceptually different physical scenarios due to lacking of a clear physical guiding principle which mass is the relevant one and due to the fact that it is not clear how the flavor wave-function relates to the spatial wave-function. There seems to be no principal contradiction. However, a nonlinear extension of the Schrodinger equation in this manner strongly depends on the relation between the flavor wave-function and spatial wave-function and its particular shape. In opposition to the Continuous Spontaneous Localization collapse models we find a change in the oscillating behavior and not in the damping of the flavor oscillation.
We modify the time dependent Schrodinger-Newton equation by using a potential for a solid sphere suggested by Jaaskelainen (Jaaskelainen 2012 Phys. Rev. A 86 052105) as well as a hollow-sphere potential. Compared to our recent paper (Giulini and Gro{
ss}ardt 2011 Class. Quantum Grav. 28 195026) where a single point-particle, i.e. a Coulomb potential, was considered this has been suggested to be a more realistic model for a molecule. Surprisingly, compared to our previous results, inhibitions of dispersion of a Gaussian wave packet occur at even smaller masses for the solid-sphere potential, given that the width of the wave packet is not exceeded by the radius of the sphere.
In this paper we show that the Schrodinger-Newton equation for spherically symmetric gravitational fields can be derived in a WKB-like expansion in 1/c from the Einstein-Klein-Gordon and Einstein-Dirac system.
We re-consider the time dependent Schrodinger-Newton equation as a model for the self-gravitational interaction of a quantum system. We numerically locate the onset of gravitationally induced inhibitions of dispersion of Gaussian wave packets and fin
d them to occur at mass values more than 6 orders of magnitude higher than reported by Salzman and Carlip (2006, 2008), namely at about $10^{10},mathrm{u}$. This fits much better to simple analytical estimates but unfortunately also questions the experimental realisability of the proposed laboratory test of quantum gravity in the foreseeable future, not just because of large masses, but also because of the need to provide sufficiently long coherence times.
