This article studies a bouncing ball model in which a discontinuous borderline oscillates so that the ball periodically transfers with in a semi-cycle or an even smaller one.For the two extremities,i.e.the conservative motion of the classic particle inside the fixed semi-cycle or the smaller poolroom,the numerical investigation shows that typical conservative features are displayed.The quantum correspondence for these two situations is also numerically studied.The two characteristic evolution features of the classic borderline oscillating model are reported.
This article studies a bouncing ball model in which a discontinuous borderline oscillates so that the ball periodically transfers with in a semi-cycle or an even smaller one.For the two extremities,i.e.the conservative motion of the classic particle inside the fixed semi-cycle or the smaller poolroom,the numerical investigation shows that typical conservative features are displayed.The quantum correspondence for these two situations is also numerically studied.The two characteristic evolution features of the classic borderline oscillating model are reported.
Quantum scars are tracks of enhanced probability densities along unstable classical periodic orbits in quantum chaotic systems. They are a visual representation of classical-quantum correspondence as well as an example of quantum suppression of chaos. A new type of quantum scarring was recently discovered in radially symmetric two-dimensional potential wells perturbed by localized impurities. These perturbation-induced scars show potential to be utilized in controlled applications in classically chaotic quantum systems due to the ability to efficiently propagate wave packets along the scars. In this thesis, the focus is on special linearly shaped perturbation-induced quantum scars which correspond to the classical motion of a ball bouncing between two walls. These bouncing ball quantum scars have potential to be used as efficient channels in quantum transport, but their behavior is still mostly unexplored. This thesis studies how bouncing ball scars could be controlled by changing the perturbation parameters and how stable these scars are against external noise. Both of these features are relevant when utilization in experimental applications is considered. Bouncing ball scars are studied in a radially symmetric two-dimensional (2D) potential well perturbed by a singular Gaussian-like potential bump or a dip, which are placed into the same location. For the stability calculations, randomly placed smaller Gaussian bumps are also placed into the potential with a constant uniform mean density in addition to the main perturbation. Imaginary time propagation algorithm is then applied to solve the 2D Schrödinger equation in our model systems. Bouncing ball scars are detected from the solved eigenstates by integrating the probability densities of the states inside narrow detection boxes. The results indicate that the singular potential bump and the dip affect the same states of the clean system in different ways. The bump seems to pin the bouncing ball scars onto itself, when the dip causes the scars to avoid it. This means that the different perturbation types have the opposite scar orientation preferences. The abundance of the bouncing ball scars seems to also be connected to the perturbation parameters, and the results indicate that the parameters could be optimized in order to maximize the abundance. The stability results show impressive robustness against external noise. The number of bouncing ball scars remain nearly constant with weaker noise and even when the noise bumps were 10% from the main perturbation size, a notable amount of bouncing ball scars persist. It also seems that the scars retain their orientations even in the noisy systems.
Abstract We review the dynamics of a bouncing ball, both from the classical and the quantum points of view, studying the role of its inertial and gravitational masses. We analyze the behavior of quantum particles of different masses, embedded in a uniform gravitational field. We show that the bouncing ball is a useful didactic example to show that the coupling between rest mass and gravity cannot be avoided at the quantum level and that the equivalence principle must be reformulated in quantum mechanics. Galileo’s experiment of falling bodies can be still satisfied if one considers only the average quantum values of observables.
Abstract 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.
Quantum Chamber Bouncing Ball: A Transcendental Data Transmission Model presents a novel theoretical framework for a quantum-confined bouncing ball system that exhibits equidistant data transmission to its point of origin. The core innovation is a transcendental amplitude recurrence relation governed by the composite factor 8π² ≈ 78.96, where each successive bounce doubles the previous amplitude and elevates it by the squared periodic conversion of π. The ball's center of mass executes sinusoidal radial oscillations superimposed on a rotating angular trajectory within a spherical quantum chamber of radius R, while data transmission is modeled as a spherical delta-distribution wavefront whose radius equals the instantaneous distance of the ball from the origin—ensuring the origin always lies on the data wavefront. The paper derives the corresponding quantum wavefunction, analyzes the energy scaling behavior (Eₙ ∝ (8π²)²ⁿ), and provides a complete computational simulation with four-panel visualization outputs including 3D trajectory, radial distance, semi-log amplitude recurrence, and equidistant data wavefront verification. The document includes three figures: Figure 1 presents the four-panel simulation output confirming all theoretical predictions to machine precision; Figure 2 visualizes the quantum wavefunction phase evolution modulo 2π, revealing the characteristic sawtooth pattern with increasing oscillation density reflecting the growing energy eigenvalues across eight bounce periods; and Figure 3 illustrates the alternative hyper-exponential formulation hₙ = (2hₙ₋₁)⁴π², demonstrating runaway amplification that exceeds 10¹⁰⁰⁰⁰ by the third bounce—confirming this interpretation as a purely mathematical construct with no physical realizability. The paper concludes with discussions on physical interpretation, energy considerations in open quantum systems, limitations including relativistic effects, and potential applications in quantum sensor networks, cavity QED, and transcendental oscillator design.
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.
Summary It is argued that quantum mechanics is a better theory than classical mechanics because it gives all the classical results and in addition explains new phenomena that classical mechanics cannot convincingly explain. However, little time seems to be spent in introductory courses on quantum mechanics to show that one can indeed obtain even the simplest classical results. In this paper we discuss two simple classical phenomena, a ball bouncing against a wall and a ball bouncing backwards and forwards inside a box. By explicitly finding the square integrable solutions of the time‐dependent Schrodinger equation we show that there is an analogous quantum mechanical bouncing process. It is possible to find explicit solutions near the classical limit since there is negligible dispersion of the wave functions, a fact that enables the integrals over the improper eigenfunctions to be evaluated with little error.