Single particle Schrödinger equation with gravitational self-interaction

Abstract

Following suggestions by several authors that the gravity may play a role in the collapse of the wavefunction during the measurement process, we have made a study of the Schrödinger–Newton Equation, which is a single particle equation, in which the degrees of freedom of the associated gravitational field of the particle are incorporated in an averaged self-consistent manner through the addition of a gravitational self-potential. We show that there exists a class of stationary self-bound solutions whose energy eigenvalues can be determined exactly using an asymptotic method. Since this equation has also been investigated in other contexts like plasma physics and astrophysics, our solutions are of larger interest. This analysis provides us with a length scale within which the predictions of standard quantum mechanics are valid, but beyond which the gravitational effects dominate. These effects do not permit spatial superpositions of wavefunction beyond this scale, leading toa possible quantum to classical transition dependent on the mass of the particle. We find a limiting mass, that is effectively the Planck mass, above which this equation may not be valid.