Schrödinger–Newton equations

The Schrödinger–Newton equations are modifications of the Schrödinger equation, proposed by Roger Penrose, that mathematically describe the basis states involved in a gravitationally-induced wavefunction collapse scheme:

: $ihbarfrac\{partialPsi\}\{partial\; t\}\; =\; -frac\{hbar^2\}\{2m\}\; abla\; ^2Psi+\; V\; Psi\; +\; mPhi\; Psi$

:$abla^2\; Phi\; =\; 4pi\; G\; m\; |Psi|^2$

where $Phi$ is a quasi-Newtonian potential given by

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

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.

References

* [http://arxiv.org/abs/math-ph/0509066 Lie point symmetries and the geodesic approximation for the Schrödinger-Newton equations]
* [http://eprints.maths.ox.ac.uk/archive/00000041/01/harrison.pdf A numerical study of the Schrodinger-Newton equations] A dissertation (note it's a pdf file).
* [http://www.dirac.org/physics/dissertation/dissertation.ps Investigation of the Time Dependent Schrodinger-Newton Equation] A dissertation (note it's post-script).
* [http://www.snarkpit.net/pits/leperous/pages/dissertation.pdf The Schrodinger-Newton Equations] Another Dissertation (note it's a pdf file).