# Schrödinger–Newton equations

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The Schrödinger–Newton equations are modifications of the Schrödinger equation and derived from Gauss' law for gravity. These equations were first considered by Runi and Bonazzola[1] in the theory of self-gravitating boson stars.

With self-gravitating boson stars, the Schrödinger–Newton equations are the non-relativistic limit of the governing Klein–Gordon equations. Runi and Bonazzola considered the problem of finding stationary boson-stars in the non-relativistic spherically symmetric case, which corresponds to finding stationary spherically symmetric solutions of the Schrödinger–Newton equations.

But the Schrödinger–Newton equation has importance for the nonrelativistic limit of quantum gravity,[2] and as a way we can explain gravity causing quantum wave function collapse.

## Overview

We consider the usual Schrödinger equation with the gravitational potential included

$i\hbar \frac{\partial\Psi}{\partial t} = -\frac{\hbar^2}{2m} \nabla ^2 \Psi + V \Psi + m \Phi \Psi$

where the gravitational potential $\Phi$ satisfies the field equation

$\nabla^2 \Phi = 4 \pi G \rho |\Psi|^2 = 4\pi G\langle\rho\rangle$

where $\rho$ is the classical mass density.

After some calculation, we find the gravitational potential can be written down as

$\Phi (\mathbf{x},t) = -G m \int_{}^{} \frac{ | \Psi(\mathbf{y},t) |^2}{|\mathbf{x} - \mathbf{y}|} \, d^3 \mathbf{y}$

It can be shown that these equations conserve probability, momentum etc., as the Schrödinger equation does.

Their Lie point symmetries are rotations, translations, scalings, a phase change in time, and a Galilean transformation of sorts that looks like the equivalence principle at work.[3]

## Quantum wave function collapse

The idea gravity causes (or somehow influences) wavefunction collapse using the Schrödinger–Newton equation dates back to the 1980s.[4]

Roger Penrose proposed that the Schrödinger–Newton equations mathematically describe the basis states involved in a gravitationally-induced wavefunction collapse scheme.[5] Penrose suggested that a superposition of two or more quantum states, which have a significant amount of mass displacement between the states, ought to be unstable and reduce to one of the states within a finite time. He hypothesises there exists a "preferred" set of states which could collapse no further, specifically the Stationary states.

K.R.W. Jones considered an extension of this proposal to gravitational collapse of an environmentally decohered quantum state. The interpretation differs in that he advocated this as a completion of the Schrödinger interpretation of a material wavefunction. The role of gravity in the scheme of Jones is to suppress macroscopic dispersion.

However, each of these proposals involves the same Schrödinger–Newton system of equations.

## Notes

1. ^ See Runi and Bonazzola "Systems of Self-Gravitating Particles in General Relativity and the Concept of an Equation of State".
2. ^ See Domenico Giulini, André Großardt, "The Schrödinger–Newton equation as non-relativistic limit of self-gravitating Klein–Gordon and Dirac fields"
3. ^ See Robertshaw and Tod, specifically section 2.2
4. ^ See Diósi.
5. ^ See Penrose 1996 and 1998.

## References

• L. Diósi, "Gravitation and quantum-mechanical localization of macro-objects", Physics Letters A 105 4–5 (1984) 199–202 doi:10.1016/0375-9601(84)90397-9
• Roger Penrose, "On Gravity's Role in Quantum State Reduction", General Relativity and Gravitation 28 5 (1996) 581-600. DOI:10.1007/BF02105068
• Roger Penrose, "Quantum computation, entanglement and state reduction", Phil. Trans. R. Soc. Lond. A 356 no. 1743 (1998) 1927-1939. DOI:10.1098/rsta.1998.0256
• P.J. Salzman, S. Carlip, "A possible experimental test of quantized gravity" arXiv:gr-qc/0606120
• Domenico Giulini, André Großardt, "The Schrödinger–Newton equation as non-relativistic limit of self-gravitating Klein–Gordon and Dirac fields" arXiv:1206.4250 [gr-qc]
• Domenico Giulini, André Großardt, "Gravitationally induced inhibitions of dispersion according to the Schrödinger–Newton Equation" Class. Quantum Grav. 28 (2011) 195026. Eprint arXiv:1105.1921 [gr-qc]
• R. Harrison "A numerical study of the Schrödinger–Newton equations", Ph.D Thesis online.
• R. Harrison, I. Moroz, K.P. Tod, "A numerical study of the Schrödinger–Newton equation 1: Perturbing the spherically-symmetric stationary states" arXiv:math-ph/0208045
• R. Harrison, I. Moroz, K.P. Tod, "A numerical study of the Schrödinger–Newton equation 2: the time-dependent problem" arXiv:math-ph/0208046
• Oliver Robertshaw, Paul Tod, "Lie point symmetries and the geodesic approximation for the Schrödinger–Newton equations" arXiv:math-ph/0509066
• R.Runi, S.Bonazzola, "Systems of Self-Gravitating Particles in General Relativity and the concept of an Equation of State", Phys.Rev. 187 5 (1969) 1767-1783. doi:10.1103/PhysRev.187.1767
• J. R. van Meter, "Schrödinger–Newton 'collapse' of the wave function", Class.Quant.Grav. 28 21 (2011) 215013. Eprint arXiv:1105.1579 [quant-ph]
• K. R. W. Jones, "Newtonian Quantum Gravity", Aust. J. of Phys. 48 6 (1995) 1055-1082. Eprint arxiv:9507001 [quant-ph]