Schrödinger-Newton equations

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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:

i\hbar\frac{\partial\Psi}{\partial t} = -\frac{\hbar^2}{2m} \nabla ^2\Psi+ V \Psi + m\Phi \Psi
\nabla^2 \Phi = 4\pi G m |\Psi|^2

where Φ is a quasi-Newtonian potential given by

\Phi (\mathbf{x},t) = -G \int_{}^{} \frac{ m | \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.

[edit] References


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