Schrödinger–Newton equations

The Schrödinger–Newton equations are modifications of the Schrödinger equation and derived from Gauss' law for gravity, proposed by Roger Penrose in The Road to Reality, that mathematically describe the basis states involved in a gravitationally-induced wavefunction collapse scheme:

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

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

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

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

and $\rho$ is the classical mass density. It can be shown that these equations conserve probability, momentum etc., as the Schrödinger equation does.