# Effective Quantum Mechanics of a falling particle

## TL;DR

This paper applies a semiclassical momentous effective quantum mechanics approach to analyze a quantum particle falling in a gravitational field, revealing that position dispersion remains near the classical trajectory under certain conditions.

## Contribution

It demonstrates that the momentous effective quantum mechanics formalism can be exactly solved for the falling particle problem, providing insights into quantum-classical correspondence.

## Key findings

- Position dispersion stays close to the classical path.
- The formalism allows for exact solutions without truncation.
- Quantum effects are captured through dispersions in a dynamical system.

## Abstract

We analyze the problem of one dimensional quantum particle falling in a constant gravitational field, also known as the {\it bouncing ball}, employing a semiclassical approach known as momentous effective quantum mechanics. In this formalism the quantum evolution is described through a dynamical system of infinite dimension for the position, the momentum and all dispersions. Usually the system is truncated to have a finite dimensional one, however, in this case equations of motion decouple and the system can be solved. For a specific set of initial conditions we find that the time dependent dispersion in position is always around the classical trajectory.