Effective dynamics of the quantum falling particle

We analyze the problem of a quantum particle falling under the influence of a one-dimensional constant gravitational field, also known as the bouncing ball, employing a semiclassical approach for the effective equations of motion for the quantum system. In this formalism, the quantum evolution is described through a dynamical system of infinite dimensions for the position, the momentum, and all dispersions. Usually, the system is truncated to reduce it to a finite-dimensional one; however, in this case, equations of motion decouple and the system can be solved exactly. For a specific set of initial conditions, we find that the time-dependent dispersion in position follows the classical trajectory; however, for large times, it grows enough to allow a non-classical behavior for the rebounds. We also propose the study of an effective potential in terms of a pair of canonical variables for dispersions.