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.

我们分析了量子粒子在一维恒定引力场（也称为弹跳球）的影响下掉落的问题，对量子系统的有效运动方程采用半经典方法。在这种形式主义中，量子演化通过位置，动量和所有色散的无穷大动力学系统来描述。通常，系统会被截断以将其缩减为有限维的系统。但是，在这种情况下，运动方程解耦，并且系统可以精确求解。对于一组特定的初始条件，我们发现位置随时间变化的离散遵循经典轨迹。但是，在很大的时间内，它的增长足以使反弹具有非经典的表现。我们还建议根据一对典型的色散变量研究有效潜力。