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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. 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. Later on it was proposed as a model to explain the quantum wave function collapse by Diósi and 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 gravitational potential : where V is an ordinary potential and the gravitational potential satisfies the Poisson equation : Because of the back coupling of the wave-function into the potential it is a nonlinear system. The integro-differential form of the equation is : It is obtained from the above system of equations by integration of the Poisson equation under the assumption that the potential must vanish at infinity. Mathematically, the Schrödinger–Newton equation is a special case of the Hartree equation for n = 2. The equation retains most of the properties of the linear Schrödinger equation. In particular it is invariant under constant phase shifts, leading to conservation of probability, and it exhibits full Galilei invariance. In addition to these symmetries, a simultaneous transformation : maps solutions of the Schrödinger–Newton equation to solutions. The stationary equation, which can be obtained in the usual manner via a separation of variables, possesses an infinite family of normalisable solutions of which only the stationary ground state is stable. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Schrödinger–Newton equation」の詳細全文を読む スポンサード リンク
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