Gravitational self-localization for spherical masses

In this work, I consider the center-of-mass wave function for a homogenous sphere under the influence of the self-interaction due to Newtonian gravity. I solve for the ground state numerically and calculate the average radius as a measure of its size. For small masses, $M\lesssim {10}^{-17}$ kg, the radial size is independent of density, and the ground state extends beyond the extent of the sphere. For masses larger than this, the ground state is contained within the sphere and to a good approximation given by the solution for an effective radial harmonic-oscillator potential. This work thus determines the limits of applicability of the point-mass Newton Schrödinger equations for spherical masses. In addition, I calculate the fringe visibility for matter-wave interferometry and find that in the low-mass case, interferometry can in principle be performed, whereas for the latter case, it becomes impossible. Based on this, I discuss this transition as a possible boundary for the quantum-classical crossover, independent of the usually evoked environmental decoherence. The two regimes meet at sphere sizes $R\approx {10}^{-7}$ m, and the density of the material causes only minor variations in this value.

Figure 1

Wave function $\Psi \left(r\right)$, shown shaded, and potential for a sphere of osmium with $M=2\times {10}^{-17}$ kg. The bare potential given by Eq. (10) divided by $V_0=G{M}^2/R$, as well as the ground-state wave function for the harmonic-oscillator approximation, are also shown, both with dashed lines. It can be seen that the effective potential is changed by the averaging and becomes less deep, while the ground state becomes less localized.

Figure 2

Average radius vs mass for the least (lithium) and most (osmium) dense solid materials is shown by solid lines. Also, the average radii for the harmonic-oscillator approximation, given by Eq. (14), and for the point source, given by Eq. (17), are shown by dashed lines. Curves are given for both lithium and osmium, as labeled.

Figure 3

Visibility as a function of mass and grating period for osmium. The visibility is seen to decrease rapidly for grating periods larger than the average radius.