From Newton to Schrödinger and Beyond

Daniel Greiner, Günter Wunner

We point out that quantum defect theory is the appropriate framework to explain and understand the behavior of the solutions of the Newton-Schr\"odinger equation. We find that, beyond ordinary quantum defect theory, the nonlinearity of the equation induces novel features, in particular a strong state dependence of the quantum defects. We show how this can be compensated by a rescaling of the energy unit.

Emanullah Hızel, Nurettin Cenk Turgay, Bahri Guldoùgan

Schrödinger-Newton (SN) equations are modifications of the Schrödinger equation proposed by Roger Penrose involved in a gravitationally-induced wave function collapse scheme. In this paper, the three dimensional time independent Schrödinger-Newton (SN) equation is analyzed. The Lie group formalism is applied to deduce symmetries of the time indepen-dent SN equation for three dimensional case. One and two dimensional optimal systems and reduced equations are obtained.

Kryukov, Alexey A

Quantum observables can be identified with vector fields on the sphere of normalized states. The resulting {\it vector representation} is used in the paper to undertake a simultaneous treatment of macroscopic and microscopic bodies in quantum mechanics. Components of the velocity and acceleration of state under Schr{\"o}dinger evolution are given a clear physical interpretation. Solutions to Schr{\"o}dinger and Newton equations are shown to be related beyond the Ehrenfest results on the motion of averages. A formula relating the normal probability distribution and the Born rule is found.