Studies of the Schroedinger-Newton Equations in D Dimensions

Details Studies of the Schroedinger-Newton Equations in D Dimensions Studies of the Schroedinger-Newton Equations in D Dimensions

2000-11-01

arxiv.org/abs/gr-qc/0011004v1

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

Latex, 18 pages, 11 figures

Scientific paper

We investigate a $D$ dimensional generalization of the Schroedinger-Newton equations, which purport to describe quantum state reduction as resulting from gravitational effects. For a single particle, the system is a combination of the Schroedinger and Poisson equations modified so that the probability density of the wavefunction is the source of the potential in the Schroedinger equation. For spherically symmetric wavefunctions, a discrete set of energy eigenvalue solutions emerges for dimensions $D<6$, accumulating at D=6. Invoking Heisenberg's uncertainty principle to assign timescales of collapse correspoding to each energy eigenvalue, we find that these timescales may vary by many orders of magnitude depending on dimension. For example, the time taken for the wavefunction of a free neutron in a spherically symmetric state to collapse is many orders of magnitude longer than the age of the universe, whereas for one confined to a box of picometer-sized cross-sectional dimensions the collapse time is about two weeks.

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