Schrödinger–Newton equation

The inclusion of a self-interaction term represents a fundamental alteration of quantum mechanics.The Schrödinger–Newton equation was first considered by Ruffini and Bonazzola[2] in connection with self-gravitating boson stars.[3] 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.[4] Later on it was proposed as a model to explain the quantum wave function collapse by Lajos Diósi[5] and Roger Penrose,[6][7][8] from whom the name "Schrödinger–Newton equation" originates.In this context, matter has quantum properties, while gravity remains classical even at the fundamental level.[9] 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.representing the interaction of the particle with its own gravitational field, satisfies the Poisson equation[13][14][15] 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.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.contains all the mutual linear interactions, e.g. electrodynamical Coulomb interactions, while the gravitational-potential term is based on the assumption that all particles perceive the same gravitational potential generated by all the marginal distributions for all the particles together.In the aforementioned approximation, the centre-of-mass wave-function satisfies the following nonlinear Schrödinger equation:[18] In the limiting case of a wide wave-function, i.e. where the width of the centre-of-mass distribution is large compared to the size of the considered object, the centre-of-mass motion is approximated well by the Schrödinger–Newton equation for a single particle.The opposite case of a narrow wave-function can be approximated by a harmonic-oscillator potential, where the Schrödinger–Newton dynamics leads to a rotation in phase space.This is true in general: nonlinear Hartree equations never have an influence 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.of this peak probability equal to the acceleration due to Newtonian gravity:The idea that gravity causes (or somehow influences) the wavefunction collapse dates back to the 1960s and was originally proposed by Károlyházy.[5] There the equation provides an estimation for the "line of demarcation" between microscopic (quantum) and macroscopic (classical) objects.Roger Penrose proposed that the Schrödinger–Newton equation mathematically describes the basis states involved in a gravitationally induced wavefunction collapse scheme.A macroscopic system can therefore never be in a spatial superposition, since the nonlinear gravitational self-interaction immediately leads to a collapse to a stationary state of the Schrödinger–Newton equation.According to Penrose's idea, when a quantum particle is measured, there is an interplay of this nonlinear collapse and environmental decoherence.Three major problems occur with this interpretation of the Schrödinger–Newton equation as the cause of the wave-function collapse: First, numerical studies[12][15][1] agreeingly find that when a wave packet "collapses" to a stationary solution, a small portion of it seems to run away to infinity.This would mean that even a completely collapsed quantum system still can be found at a distant location.If the environment is taken into account, this effect might disappear and therefore not be present in the scenario described by Penrose.A good model for the collapse process also has to explain why the dot appears on different positions of the screen, with probabilities that are determined by the squared absolute-value of the wave-function.It might be possible that a model based on Penrose's idea could provide such an explanation, but there is as yet no known reason that Born's rule would naturally arise from it.Making use of the nonlocal nature of entangled quantum systems, this could be used to send signals faster than light, which is generally thought to be in contradiction with causality.It is, however, not clear whether this problem can be resolved by applying the right collapse prescription, yet to be found, consistently to the full quantum system.
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