19 June 2000 Ž . Physics Letters A 271 2000 157–166 www.elsevier.nlrlocaterpla Single particle Schrodinger equation with gravitational ¨ self-interaction Deepak Kumar a, ) , Vikram Soni b a School of Physical Sciences, Jawaharlal Nehru UniÕersity, New Delhi, 110067, India b National Physical Laboratory, K.S. Krishnan Marg, New Delhi, 110016, India Received 12 January 2000; received in revised form 4 May 2000; accepted 19 May 2000 Communicated by P.R. Holland 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 Schrodinger–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. q 2000 Elsevier Science B.V. All rights reserved. PACS: 03.65.Bz; 03.65.Ge Keywords: Quantum measurement; Gravitation; Quantum superposition; Macroscopic objects 1. Introduction In spite of the tremendous success of quantum mechanics for the past eighty years or so in explain- ing the behaviour of matter and radiation at allscales to which it can be applied, its interpretational aspects continue to puzzle us, particularly in regard to mea- ) Corresponding author. Ž . E-mail address: deepak@jnuniv.ernet.in D. Kumar . surements and extrapolation to the macroscopic scale. The process of measurement involves a collapse of the wavefunction to one of the eigenstates of the operator being measured, and is not describable within the framework of the normal unitary evolu- tion of the quantum mechanics. Further the quantum description when extrapolated to the macroscopic domain leads to paradoxical situations, which basi- cally arise when we superpose macroscopically dis- tinguishable quantum states. These two problems are related to each other, as the measurement process at 0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. Ž . PII: S0375-9601 00 00361-3