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Register a new userGravitationally induced inhibitions of dispersion according to the Schrodinger-Newton Equation
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Added by
Andr\\'e Gro{\\ss}ardt
Publication date
2011
fields
Physics
and research's language is
English
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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 find 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.
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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 study the solutions of the Dirac equation in the background of the Nutku helicoid metric. This metric has curvature singularities, which necessitates imposing a boundary to exclude this point. We use the Atiyah-Patodi-Singer non local spectral boundary conditions for both the four and the five dimensional manifolds.
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Dirac equation written on the boundary of the Nutku helicoid space consists of a system of ordinary differential equations. We tried to analyze this system and we found that it has a higher singularity than those of the Heuns equations which give the solutions of the Dirac equation in the bulk. We also lose an independent integral of motion on the boundary. This facts explain why we could not find the solution of the system on the boundary in terms of known functions. We make the stability analysis of the helicoid and catenoid cases and end up with an appendix which gives a new example where one encounters a form of the Heun equation.
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