Schrödinger–Newton equation

Summary

The Schrödinger–Newton equation, sometimes referred to as the Newton–Schrödinger or Schrödinger–Poisson equation, is a nonlinear modification of the Schrödinger equation with a Newtonian gravitational potential, where the gravitational potential emerges from the treatment of the wave function as a mass density, including a term that represents interaction of a particle with its own gravitational field. The inclusion of a self-interaction term represents a fundamental alteration of quantum mechanics. It can be written either as a single integro-differential equation or as a coupled system of a Schrödinger and a Poisson equation. In the latter case it is also referred to in the plural form. The Schrödinger–Newton equation was first considered by Ruffini and Bonazzola in connection with self-gravitating boson stars. In this context of classical general relativity it appears as the non-relativistic limit of either the Klein–Gordon equation or the Dirac equatio

Official source

https://en.wikipedia.org/wiki/Schr%C3%B6dinger%E2%80%93Newton_equation

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In the absence of a full analytical treatment of nonlinear structure formation in the universe, numerical simulations provide the critical link between the properties of the underlying model and the features of the observed structures. Currently N-body simulations are the main tool to study structure growth. We explore an alternative framework for numerical simulations of structure formation. The underlying idea is to replace the long-range gravitational force in the Vlasov-Poisson system by a purely local interaction. To this end we trade the classical phase space distribution for its quantum mechanical counterpart, the Wigner distribution function. Its dynamical equation is equivalent to the Schrödinger equation and reduces to the Vlasov equation in the formal semi-classical limit. The proposed framework allows in principle to simulate systems with arbitrary phase space distributions and could for instance be beneficial for simulations of warm dark matter, where the velocity dispersion is important. We discuss several methods to obtain a set of wavefunctions whose Wigner distribution is close to a given initial phase space distribution function. An auxiliary gauge field is introduced to mediate the gravitational interaction, thereby obtaining a local Schrödinger-Maxwell system. We also use the ideas of lattice gauge theories to obtain a fully gauge-invariant discretization of the equations of motion. Their iterative solution was implemented in a three-dimensional simulation code. We discuss its computational complexity and memory requirements. Several testbed simulations were performed with this method. We compared the gravitational collapse of a Gaussian wavefunction with an independent numerical solution of the spherically symmetric Schrödinger-Newton system. The results were found to be in good agreement. Finally, a simple example of the growth of cosmic perturbations is investigated within our framework. We conclude by outlining various possible directions to optimize and develop our method.

EPFL2011

The One-Dimensional Schrödinger–Newton Equations

Philippe Choquard, Joachim Stubbe

Springer, Springer Netherlands2007

Bound states of the Schrodinger-Newton model in low dimensions

Joachim Stubbe, Marc Vuffray

We prove the existence of quasi-stationary symmetric solutions with exactly n >= 0 zeros and uniqueness for n = 0 for the Schrodinger-Newton model in one dimension and in two dimensions along with an angular momentum m >= 0. Our result is based on an analysis of the corresponding system of second-order differential equations. (C) 2010 Elsevier Ltd. All rights reserved.

2010

EPFL2011