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 equation in a curved space-time together with the Einstein field equations. The equation also describes fuzzy dark matter and approximates classical cold dark matter described by the Vlasov–Poisson equation in the limit that the particle mass is large. Later on it was proposed as a model to explain the quantum wave function collapse by Lajos Diósi and Roger Penrose, from whom the name "Schrödinger–Newton equation" originates. In this context, matter has quantum properties, while gravity remains classical even at the fundamental level. The Schrödinger–Newton equation was therefore also suggested as a way to test the necessity of quantum gravity. In a third context, the Schrödinger–Newton equation appears as a Hartree approximation for the mutual gravitational interaction in a system of a large number of particles. In this context, a corresponding equation for the electromagnetic Coulomb interaction was suggested by Philippe Choquard at the 1976 Symposium on Coulomb Systems in Lausanne to describe one-component plasmas. Elliott H. Lieb provided the proof for the existence and uniqueness of a stationary ground state and referred to the equation as the Choquard equation.
Overview As a coupled system, the Schrödinger–Newton equations are the usual Schrödinger equation with a self-interaction gravitational potential
i ℏ ∂ Ψ ∂ t = − ℏ 2 2 M ∇ 2 Ψ + V Ψ + M Φ Ψ , {\displaystyle \mathrm {i} \hbar \ {\frac {\partial \Psi }{\ \partial t\ }}=-{\frac {\ \hbar ^{2}}{\ 2\ M\ }}\ \nabla ^{2}\Psi \;+\;V\ \Psi \;+\;M\ \Phi \ \Psi \ ,}
where V is an ordinary potential, and the gravitational potential Φ , {\displaystyle \ \Phi \ ,} representing the interaction of the particle with its own gravitational field, satisfies the Poisson equation
∇ 2 Φ = 4 π G M | Ψ | 2 . {\displaystyle \ \nabla ^{2}\Phi =4\pi \ G\ M\ |\Psi |^{2}~.}
Because of the back coupling of the wave-function into the potential, it is a nonlinear system. Replacing Φ {\displaystyle \ \Phi \ } with the solution to the Poisson equation produces the integro-differential form of the Schrödinger–Newton equation for a particle with mass M:
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