Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Classical and Quantum Gravity

Spherically-symmetric solutions of the Schrödinger-Newton equations

Irene M. Moroz, Roger Penrose, Paul Tod

Published Sep 1, 1998• 458 citations
Publisher

Abstract

As part of a programme in which quantum state reduction is understood as a gravitational phenomenon, we consider the Schrödinger-Newton equations. For a single particle, this is a coupled system consisting of the Schrödinger equation for the particle moving in its own gravitational field, where this is generated by its own probability density via the Poisson equation. Restricting to the spherically-symmetric case, we find numerical evidence for a discrete family of solutions, everywhere regular, and with normalizable wavefunctions. The solutions are labelled by the non-negative integers, the n th solution having n zeros in the wavefunction. Furthermore, these are the only globally defined solutions. Analytical support is provided for some of the features found numerically.

DOI
10.1088/0264-9381/15/9/019

Related papers

Studies of the Schroedinger-Newton Equations in D Dimensions

Roger G. Melko, Robert B. Mann

We investigate a $D$ dimensional generalization of the Schroedinger-Newton equations, which purport to describe quantum state reduction as resulting from gravitational effects. For a single particle, the system is a combination of the Schroedinger and Poisson equations modified so that the probability density of the wavefunction is the source of the potential in the Schroedinger equation. For spherically symmetric wavefunctions, a discrete set of energy eigenvalue solutions emerges for dimensions $D<6$, accumulating at D=6. Invoking Heisenberg's uncertainty principle to assign timescales of collapse correspoding to each energy eigenvalue, we find that these timescales may vary by many orders of magnitude depending on dimension. For example, the time taken for the wavefunction of a free neutron in a spherically symmetric state to collapse is many orders of magnitude longer than the age of the universe, whereas for one confined to a box of picometer-sized cross-sectional dimensions the collapse time is about two weeks.

arXiv:gr-qc/0011004•Nov 1, 2000•1 citation

A numerical study of the Schrödinger-Newton equations

R. Harrison

The Schrödinger-Newton (S-N) equations were proposed by Penrose [18] as a model for gravitational collapse of the wave-function. The potential in the Schrödinger equation is the gravity due to the density of $|\psi|^2$, where $\psi$ is the wave-function. As with normal Quantum Mechanics the probability, momentum and angular momentum are conserved. We first consider the spherically symmetric case, here the stationary solutions have been found numerically by Moroz et al [15] and Jones et al [3]. The ground state which has the lowest energy has no zeros. The higher states are such that the $(n+1)$th state has $n$ zeros. We consider the linear stability problem for the stationary states, which we numerically solve using spectral methods. The ground state is linearly stable since it has only imaginary eigenvalues. The higher states are linearly unstable having imaginary eigenvalues except for $n$ quadruples of complex eigenvalues for the $(n+1)$th state, where a quadruple consists of $\{\lambda,\bar{\lambda},-\lambda,-\bar{\lambda}\}$. Next we consider the nonlinear evolution, using a method involving an iteration to calculate the potential at the next time step and Crank-Nicolson to evolve the Schrödinger equation. To absorb scatter we use a sponge factor which reduces the reflection back from the outer boundary condition and we show that the numerical evolution converges for different mesh sizes and time steps. Evolution of the ground state shows it is stable and added perturbations oscillate at frequencies determined by the linear perturbation theory. The higher states are shown to be unstable, emitting scatter and leaving a rescaled ground state. The rate at which they decay is controlled by the complex eigenvalues of the linear perturbation. Next we consider adding another dimension in two different ways: by considering the axisymmetric case and the 2-D equations. The stationary solutions are found. We modify the evolution method and find that the higher states are unstable. In 2-D case we consider rigidly rotationing solutions and show they exist and are unstable.

•Jan 1, 2001•0 citations

Relativistic generalization of the Schrödinger-Newton model for the wavefunction reduction

Vladimir V. Kassandrov, Nina V. Markova

We consider the model of the self-gravity driven spontaneous wavefunction reduction proposed by L. Diosi, R. Penrose et al. and based on a self-consistent system of Schrödinger and Poisson equations. An analogous system of coupled Dirac and Maxwell-like equations is proposed as a relativization. Regular solutions to the latter form a discrete spectrum in which all the “active” gravitational masses are always positive, and approximately equal to inertial masses and to the mass [Formula: see text] of the quanta of Dirac field up to the corrections of order [Formula: see text]. Here [Formula: see text] is the gravitational analogue of the fine structure constant negligibly small for nucleons. In the limit [Formula: see text] the model reduces back to the nonrelativistic Schrödinger-Newton one. The equivalence principle is fulfilled with an extremely high precision. The above solutions correspond to various states of the same (free) particle rather than to different particles. These states possess a negligibly small difference in characteristics but essentially differ in the widths of the wavefunctions. For the ground state the latter is [Formula: see text] times larger the Compton length, so that a nucleon cannot be sufficiently localized to model the reduction process.

arXiv:2010.03971•International Journal of Modern Physics A•Jan 24, 2020•2 citations

Schrödinger–Newton ‘collapse’ of the wavefunction

James R. van Meter

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.

arXiv:1105.1579•Classical and Quantum Gravity•Sep 22, 2011•52 citations

The dynamics of the Schrödinger–Newton system with self-field coupling

Joel Franklin, Yongwei Guo, Katherine Newton +1 more

Abstract We probe the dynamics of a modified form of the Schrödinger–Newton (SN) system of gravity coupled to single particle quantum mechanics. At the masses of interest here, the ones associated with the onset of ‘collapse’ (where the gravitational attraction is competitive with the quantum mechanical dissipation), we show that the Schrödinger ground state energies match the Dirac ones with an error of . At the Planck mass scale, we predict the critical mass at which a potential collapse could occur for the self-coupled gravitational case, Planck mass, and show that gravitational attraction opposes Gaussian spreading at around this value, which is a factor of two higher than the one predicted (and verified) for the SN system. Unlike the SN dynamics, we do not find that the self-coupled case tends to decay towards its ground state; there is no collapse in this case.

arXiv:1603.03380•Classical and Quantum Gravity•Mar 1, 2016•3 citations

The Schrödinger–Newton equation as a possible generator of quantum state reduction

Jasper van Wezel, Jeroen van den Brink

It has been suggested by Diósi and Penrose that the occurrence of quantum state reduction in macroscopic objects is related to a manifestation of gravitational effects in quantum mechanics. Although within Penrose's framework the dynamics of the quantum state reduction is not prescribed, it was suggested that the so-called Schrödinger–Newton equation can be used to at least identify the resulting classical end states. Here we analyse the extent to which the Schrödinger–Newton equation can be used as a model to generate a full, time-dependent description of the quantum state reduction process. We find that when supplied with an imaginary gravitational potential, the Schrödinger–Newton equation offers a rationalization for some of the hitherto unexplained characteristics of quantum state reduction. The description remains incomplete however, because it is unclear how to fully recover Born's rule.

arXiv:0803.4488•The Philosophical Magazine A Journal of Theoretical Experimental and Applied Physics•Apr 11, 2008•12 citations
As part of a programme in which quantum state reduction is understood as a gravitational phenomenon, we consider the Schrödinger-Newton equations. For a single particle, this is a coupled system consisting of the Schrödinger equation for the particle moving in its own gravitational field, where this is generated by its own probability density via the Poisson equation. Restricting to the spherically-symmetric case, we find numerical evidence for a discrete family of solutions, everywhere regular, and with normalizable wavefunctions. The solutions are labelled by the non-negative integers, the n th solution having n zeros in the wavefunction. Furthermore, these are the only globally defined solutions. Analytical support is provided for some of the features found numerically.
We investigate a DDD dimensional generalization of the Schroedinger-Newton equations, which purport to describe quantum state reduction as resulting from gravitational effects. For a single particle, the system is a combination of the Schroedinger and Poisson equations modified so that the probability density of the wavefunction is the source of the potential in the Schroedinger equation. For spherically symmetric wavefunctions, a discrete set of energy eigenvalue solutions emerges for dimensions D<6D<6D<6, accumulating at D=6. Invoking Heisenberg's uncertainty principle to assign timescales of collapse correspoding to each energy eigenvalue, we find that these timescales may vary by many orders of magnitude depending on dimension. For example, the time taken for the wavefunction of a free neutron in a spherically symmetric state to collapse is many orders of magnitude longer than the age of the universe, whereas for one confined to a box of picometer-sized cross-sectional dimensions the collapse time is about two weeks.
The Schrödinger-Newton (S-N) equations were proposed by Penrose [18] as a model for gravitational collapse of the wave-function. The potential in the Schrödinger equation is the gravity due to the density of ∣ψ∣2|\psi|^2∣ψ∣2, where ψ\psiψ is the wave-function. As with normal Quantum Mechanics the probability, momentum and angular momentum are conserved. We first consider the spherically symmetric case, here the stationary solutions have been found numerically by Moroz et al [15] and Jones et al [3]. The ground state which has the lowest energy has no zeros. The higher states are such that the (n+1)(n+1)(n+1)th state has nnn zeros. We consider the linear stability problem for the stationary states, which we numerically solve using spectral methods. The ground state is linearly stable since it has only imaginary eigenvalues. The higher states are linearly unstable having imaginary eigenvalues except for nnn quadruples of complex eigenvalues for the (n+1)(n+1)(n+1)th state, where a quadruple consists of {λ,λˉ,−λ,−λˉ}\{\lambda,\bar{\lambda},-\lambda,-\bar{\lambda}\}{λ,λˉ,−λ,−λˉ}. Next we consider the nonlinear evolution, using a method involving an iteration to calculate the potential at the next time step and Crank-Nicolson to evolve the Schrödinger equation. To absorb scatter we use a sponge factor which reduces the reflection back from the outer boundary condition and we show that the numerical evolution converges for different mesh sizes and time steps. Evolution of the ground state shows it is stable and added perturbations oscillate at frequencies determined by the linear perturbation theory. The higher states are shown to be unstable, emitting scatter and leaving a rescaled ground state. The rate at which they decay is controlled by the complex eigenvalues of the linear perturbation. Next we consider adding another dimension in two different ways: by considering the axisymmetric case and the 2-D equations. The stationary solutions are found. We modify the evolution method and find that the higher states are unstable. In 2-D case we consider rigidly rotationing solutions and show they exist and are unstable.
We consider the model of the self-gravity driven spontaneous wavefunction reduction proposed by L. Diosi, R. Penrose et al. and based on a self-consistent system of Schrödinger and Poisson equations. An analogous system of coupled Dirac and Maxwell-like equations is proposed as a relativization. Regular solutions to the latter form a discrete spectrum in which all the “active” gravitational masses are always positive, and approximately equal to inertial masses and to the mass [Formula: see text] of the quanta of Dirac field up to the corrections of order [Formula: see text]. Here [Formula: see text] is the gravitational analogue of the fine structure constant negligibly small for nucleons. In the limit [Formula: see text] the model reduces back to the nonrelativistic Schrödinger-Newton one. The equivalence principle is fulfilled with an extremely high precision. The above solutions correspond to various states of the same (free) particle rather than to different particles. These states possess a negligibly small difference in characteristics but essentially differ in the widths of the wavefunctions. For the ground state the latter is [Formula: see text] times larger the Compton length, so that a nucleon cannot be sufficiently localized to model the reduction process.
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.
Abstract We probe the dynamics of a modified form of the Schrödinger–Newton (SN) system of gravity coupled to single particle quantum mechanics. At the masses of interest here, the ones associated with the onset of ‘collapse’ (where the gravitational attraction is competitive with the quantum mechanical dissipation), we show that the Schrödinger ground state energies match the Dirac ones with an error of . At the Planck mass scale, we predict the critical mass at which a potential collapse could occur for the self-coupled gravitational case, Planck mass, and show that gravitational attraction opposes Gaussian spreading at around this value, which is a factor of two higher than the one predicted (and verified) for the SN system. Unlike the SN dynamics, we do not find that the self-coupled case tends to decay towards its ground state; there is no collapse in this case.
It has been suggested by Diósi and Penrose that the occurrence of quantum state reduction in macroscopic objects is related to a manifestation of gravitational effects in quantum mechanics. Although within Penrose's framework the dynamics of the quantum state reduction is not prescribed, it was suggested that the so-called Schrödinger–Newton equation can be used to at least identify the resulting classical end states. Here we analyse the extent to which the Schrödinger–Newton equation can be used as a model to generate a full, time-dependent description of the quantum state reduction process. We find that when supplied with an imaginary gravitational potential, the Schrödinger–Newton equation offers a rationalization for some of the hitherto unexplained characteristics of quantum state reduction. The description remains incomplete however, because it is unclear how to fully recover Born's rule.