Шрёдингер-Ньютон теңдеуі: кванттық механиканың бұрылмалы модификациясы

Nonlinear modification of the Schrödinger equation
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. 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.

The Schrödinger–Newton equation can be derived under the assumption that gravity remains classical, even at the fundamental level, and that the right way to couple quantum matter to gravity is by means of the semiclassical Einstein equations. In this case, a Newtonian gravitational potential term is added to the Schrödinger equation, where the source of this gravitational potential is the expectation value of the mass density operator or mass flux current. In this regard, if gravity is fundamentally classical, the Schrödinger–Newton equation is a fundamental one particle equation, which can be generalised to the case of many particles (see below). If, on the other hand, the gravitational field is quantised, the fundamental Schrödinger equation remains linear. The Schrödinger–Newton equation is then only valid as an approximation for the gravitational interaction in systems of a large number of particles and has no effect on the centre of mass.

A rough order of magnitude estimate of the regime where effects of the Schrödinger–Newton equation become relevant can be obtained by a rather simple reasoning. show that this equation gives a rather good estimate of the mass regime above which effects of the Schrödinger–Newton equation become significant. For an atom the critical width is around 1022 metres, while it is already down to 10−31 metres for a mass of one microgram. The regime where the mass is around 1010 atomic mass units while the width is of the order of micrometers is expected to allow an experimental test of the Schrödinger–Newton equation in the future. A possible candidate are interferometry experiments with heavy molecules, which currently reach masses up to 10,000 atomic mass units.

The idea that gravity causes (or somehow influences) the wavefunction collapse dates back to the 1960s and was originally proposed by Károlyházy. The Schrödinger–Newton equation was proposed in this context by Diósi. about a similar thought experiment proposed by Eppley & Hannah).