It is proposed that the Schrodinger equation for a free point particle has non-linear corrections which depend on the mass of the particle. It is assumed that the corrections become extremely small when the mass is much smaller or much larger than a critical value (the critical value being related to but smaller than Planck mass). The corrections become significant when the mass is close to this critical value and could play a role in explaining wave-function collapse. It appears that such corrections are not ruled out by present day experimental tests of the Schrodinger equation. Corrections to the energy levels of a harmonic oscillator are calculated.
It is proposed that the Schrodinger equation for a free point particle has non-linear corrections which depend on the mass of the particle. It is assumed that the corrections become extremely small when the mass is much smaller or much larger than a critical value (the critical value being related to but smaller than Planck mass). The corrections become significant when the mass is close to this critical value and could play a role in explaining wave-function collapse. It appears that such corrections are not ruled out by present day experimental tests of the Schrodinger equation. Corrections to the energy levels of a harmonic oscillator are calculated.
We present some physically interesting, in general non-stationary, one-dimensional solutions to the nonlinear phase modification of the Schrödinger equation proposed recently. The solutions include a coherent state, a phase-modified Gaussian wave packet in the potential of harmonic oscillator whose strength varies in time, a free Gaussian soliton, and a similar soliton in the potential of harmonic oscillator comoving with the soliton. The last of these solutions implies that there exist an energy level in the spectrum of harmonic oscillator which is not predicted by the linear theory. The free solitonic solution can be considered a model for a particle aspect of the wave-particle duality embodied in the quantum theory. The physical size of this particle is naturally rendered equal to its Compton wavelength in the subrelativistic framework in which the self-energy of the soliton is assumed to be equal to its rest-mass energy. The solitonic solutions exist only for the negative coupling constant for which the Gaussian wave packets must be larger than some critical finite size if their energy is to be bounded, i.e., they cannot be point-like objects.
It has been suggested that the nonlinear Schrödinger–Newton equation might approximate the coupling of quantum mechanics with gravitation, particularly in the context of the Møller–Rosenfeld semiclassical theory. Numerical results for the spherically symmetric, time-dependent, single-particle case are presented, clarifying and extending previous work on the subject. It is found that, for a particle mass greater than 1.14(ℏ 2 /( G σ)) 1/3 , a wave packet of width σ partially 'collapses' to a groundstate solution found by Moroz, Penrose and Tod, with excess probability dispersing away. However, for a mass less than 1.14(ℏ 2 /( G σ)) 1/3 , the entire wave packet appears to spread like a free particle, albeit more slowly. It is argued that, on some scales (lower than the Planck scale), this theory predicts significant deviation from conventional (linear) quantum mechanics. However, owing to the difficulty of controlling quantum coherence on one hand, and the weakness of gravity on the other, definitive experimental falsification poses a technologically formidable challenge.
There are fundamental reasons why there should exist a reformulation of quantum mechanics which does not refer to a classical space–time manifold. It follows that quantum mechanics as we know it is a limiting case of a more general nonlinear quantum theory, with the nonlinearity becoming significant at the Planck mass/energy scale. This nonlinearity is responsible for a dynamically induced collapse of the wave function, during a quantum measurement, and it hence falsifies the many-worlds interpretation of quantum mechanics. We illustrate this conclusion using a mathematical model based on a generalized Doebner–Goldin equation. The non-Hermitian part of the Hamiltonian in this norm-preserving, nonlinear, Schrödinger equation dominates during a quantum measurement, and leads to a breakdown of linear superposition.
The Schrödinger equation has long been treated as a foundational postulate of quantum mechanics. We show that it doesn't have to be. Starting from four minimal physical axioms — matter as a complex oscillation field, global phase invariance, stationary action for a local, linear Lagrangian, and the non-relativistic limit — the equation follows as a mathematical necessity, unique up to two constants fixed by correspondence with classical free-particle dynamics. A fifth statement — probability conservation — follows as a theorem, as we verify explicitly. The derivation also points to a natural scale. Comparing the quantum zero-point length of an oscillator with the correlation length of its environment defines a critical mass mc = ħ / (ω Δx_th^2). It depends on the oscillator frequency ω and the environmental length Δx_th, which are properties of the system and its environment rather than of the axioms. Below mc, zero-point motion dominates environmental noise and quantum coherence can be sustained; above it, environmental noise dominates and coherence is lost. Near mc, we model the phase as a diffusive field with local white noise. This assumed model gives, for effective dimension one, a 1/f^1.5 power spectral density, a departure from the f^-2 thermal floor, which we propose as a working hypothesis for experiments with levitated nanoparticles and cryogenic micro-resonators. These results suggest a physical criterion for Schrödinger's cat: it is not a paradox, but a limit. The cat's mass exceeds mc by many orders of magnitude, placing it deep in the classical regime without invoking an observer or multiple universes.
A nonlinear extension of Schrödinger's wave equation is proposed that ensures non-signaling by keeping linear the evolution of \textit{coordinate-diagonal} elements of the density matrix. The equation contains a negative kinetic energy term that turns spreading of wave packets into its opposite: collapsing, as some effective mass M grows beyond a universal critical mass, estimated to be about μ=2⋅10−23kg; then linear quantum kinetic energy gets negligible, which marks the quantum-classical border. Interference of large molecules is suggested for an experimental check of the proposed framework.
There are few proposals, which explicitly allow for (experimentally testable) deviations from standard quantum theory. Collapse models are among the most-widely studied proposals of this kind. The Schrödinger equation is modified by including nonlinear and stochastic terms, which describe the collapse of the wave function in space. These spontaneous collapses are rare for microscopic systems, hence their quantum properties are left almost unaltered. On the other hand, collapses become more and more frequent, the larger the object, to the point that macroscopic superpositions are rapidly suppressed. The main features of collapse models will be reviewed. An update of the most promising experimental tests will be presented.