We investigate the implication of the nonlinear and non-local multi-particle Schrödinger–Newton equation for the motion of the mass centre of an extended multi-particle object, giving self-contained and comprehensible derivations. In particular, we discuss two opposite limiting cases. In the first case, the width of the centre-of-mass wave packet is assumed much larger than the actual extent of the object, in the second case it is assumed much smaller. Both cases result in nonlinear deviations from ordinary free Schrödinger evolution for the centre of mass. On a general conceptual level we include some discussion in order to clarify the physical basis and intention for studying the Schrödinger–Newton equation.
We investigate the implication of the nonlinear and non-local multi-particle Schrödinger–Newton equation for the motion of the mass centre of an extended multi-particle object, giving self-contained and comprehensible derivations. In particular, we discuss two opposite limiting cases. In the first case, the width of the centre-of-mass wave packet is assumed much larger than the actual extent of the object, in the second case it is assumed much smaller. Both cases result in nonlinear deviations from ordinary free Schrödinger evolution for the centre of mass. On a general conceptual level we include some discussion in order to clarify the physical basis and intention for studying the Schrödinger–Newton equation.
We investigate the implication of the nonlinear and non-local multi-particle Schrodinger-Newton equation for the motion of the mass centre of an extended multi-particle object, giving self-contained and comprehensible derivations. In particular, we discuss two opposite limiting cases. In the first case, the width of the centre-of-mass wave packet is assumed much larger than the actual extent of the object, in the second case it is assumed much smaller. Both cases result in nonlinear deviations from ordinary free Schrodinger evolution for the centre of mass. On a general conceptual level we include some discussion in order to clarify the physical basis and intention for studying the Schrodinger-Newton equation.
The evolution of the center-of-mass wave function for a mesoscopic particle according to the Schr\"odinger-Newton equation can be approximated by a harmonic potential if the wave function is narrow compared to the size of the mesoscopic particle. It was noticed by Colin et al. [Phys. Rev. A 93, 062102 (2016).] that, in the regime where self-gravitational effects are weak, intermediate and wider wave functions may be approximated by a harmonic potential as well but with a width-dependent coupling, leading to a time evolution that is determined only by a differential equation for the width of a Gaussian wave function as a single parameter. Such an approximation results in considerably less computational effort in order to predict the self-gravitational effects on the wave-function dynamics. Here, we provide an alternative approach to this kind of approximation, including a rigorous derivation of the equations of motion for an initially Gaussian wave packet under the assumption that its shape is conserved. Our result deviates to some degree from the result by Colin et al. [Phys. Rev. A 93, 062102 (2016).], specifically in the limit of wide wave functions.
We examine the origin of the Newton–Schrödinger equations (NSEs) that play an important role in alternative quantum theories (AQT), macroscopic quantum mechanics and gravity-induced decoherence. We show that NSEs for individual particles do not follow from general relativity (GR) plus quantum field theory (QFT). Contrary to what is commonly assumed, the NSEs are not the weak-field (WF), non-relativistic (NR) limit of the semi-classical Einstein equation (SCE) (this nomenclature is preferred over the 'Moller–Rosenfeld equation') based on GR+QFT. The wave-function in the NSEs makes sense only as that for a mean field describing a system of N particles as , not that of a single or finite many particles. From GR+QFT the gravitational self-interaction leads to mass renormalization, not to a non-linear term in the evolution equations of some AQTs. The WF-NR limit of the gravitational interaction in GR+QFT involves no dynamics. To see the contrast, we give a derivation of the equation (i) governing the many-body wave function from GR+QFT and (ii) for the non-relativistic limit of quantum electrodynamics. They have the same structure, being linear, and very different from NSEs. Adding to this our earlier consideration that for gravitational decoherence the master equations based on GR+QFT lead to decoherence in the energy basis and not in the position basis, despite some AQTs desiring it for the 'collapse of the wave function', we conclude that the origins and consequences of NSEs are very different, and should be clearly demarcated from those of the SCE equation, the only legitimate representative of semiclassical gravity, based on GR+QFT.
In describing the motion of atoms and clusters, we face with choosing quantum mechanics or classical mechanics under different conditions. In principle, there exist two criteria for this choice, but they do contradict in some cases though they are in agreement for other cases. Actually, this problem is closely related with the effective centre-of-mass method, the underlying application of quantum mechanics. It is shown that quantum mechanics must be selected for particle's motion when the de Broglie wave length of the mass centre is larger than the particle size, and in such case the effective centre-of-mass can be used in Quantum Mechanics. In order to test this conclusion, an easy-manufactured experiment is suggested.
Abstract-- We used a nonlinear Schrodinger equation to replace the linear Schrodinger equation and to study further the natures and states of microscopic particles due to plenty of difficulties of quantum mechanics. From this investigation we find that the states and properties of microscopic particles are considerably changed relative to those in quantum mechanics. An outstanding and obvious change is that the microscopic particles have a wave-corpuscle duality in such a case. The solution of the nonlinear Schrodinger equation contains an envelop and carrier waves with determinant frequency which can propagation in medium in a certain velocity. These display the wave feature of particle. However the solutions have all a mass centre and possess a determinant size and mass, momentum and energy, which satisfy also the conservation laws of mass, momentum and energy, at the same time, they meet the collision law of classical particles. These embody the corpuscle feature of the microscopic particles. Finally we seek the reasons and roots generating these changes, which are due to the nonlinear interactions among the particles or between the particles and background field to be considered in the nonlinear Schrodinger equation. The nonlinear interactions provide a double-well potential to make the microscopic particle to be localized as a soliton, and break also through the fundamental hypothesis for the independence of Hamiltonian
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.