Schrödinger–Newton Equation with Spontaneous Wave Function Collapse

Abstract

Based on the assumption that the standard Schrödinger equation becomes gravitationally modified for massive macroscopic objects, two independent proposals have survived from the 1980s. The Schrödinger–Newton equation (1984) provides well-localized solitons for free macro-objects but lacks the mechanism of how extended wave functions collapse on solitons. The gravity-related stochastic Schrödinger equation (1989) provides the spontaneous collapse, but the resulting solitons undergo a tiny diffusion, leading to an inconvenient steady increase in the kinetic energy. We propose the stochastic Schrödinger–Newton equation, which contains the above two gravity-related modifications together. Then, the wave functions of free macroscopic bodies will gradually and stochastically collapse to solitons, which perform inertial motion without momentum diffusion: conservation of momentum and energy is restored.

Explanation of Relation between Wave Function and Probability Density Based on Quantum Mechanics in Phase Space

“…The recent researches show that the wave function as a very peculiar object is interpreted still differently by two strategies: either something really existing [11]- [16] or an abstract entity for describing the behavior of particles [11] [17] [18]. Without correct interpretation of the wave function, it is impossible to completely solve the important foundation problems of quantum mechanics including the wave function collapse [19] and quantum entanglement.…”