Trajectory of a massive localised wave function in a curved spacetime geometry

Propagation of a localised wave function of a massive scalar field is investigated in its rest frame. The complete orthogonal Hermite-Gauss basis is presented, and the Gouy phase and Rayleigh scale notions are adapted. The leading and sub-leading gravitational corrections to a localised quantum wave function propagating in a general curved spacetime geometry are calculated within the Fermi coordinates around the time-like geodesic of its rest frame, and cross-talk coefficients among the modes are derived. It is observed that spherically symmetric modes propagate along the geodesic. However, non-spherical modes are found to experience a mode-dependent residual quantum force at the sub-leading order. It is shown that the residual force does not generate an escape velocity for in-falling wave functions but leads to a mode-dependent deflection angle for the scattered ones.