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arXiv:2410.10351 · The European Physical Journal Plus

Scaled quantum theory: the bouncing ball problem

S. V. Mousavi, Salvador Miret-Artés

Published Oct 24, 2024• 3 citations
arXivHTMLPDF
DOI
10.1140/epjp/s13360-024-05741-8

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We started to investigate quantum mechanics by considering only the quantization of magnetic arrows. In our explorations we found out that the magnetic arrow had some funny properties (for example, it was possible that m x did not have a definite value), but at first it seemed that other properties, such as the position of an atom, behaved in the familiar classical way. Eventually (section 9.3) we found that it was also possible to have an atom without a definite value for its position. In this chapter, we investigate what happens when we apply quantum mechanics to a particle's position. Ball bouncing from a floor This chapter will show our framework for quantum mechanics in action, by applying it to the problem of a ball bouncing from a floor. Let us use a very fast ball, such as an electron, so that we can ignore the force of gravity. (We restrict ourselves to an electron that is moving fast on a human scale but slow compared to the speed of light, so that relativistic considerations don't come into play. Also, the magnetic arrow associated with the electron has no effect on the phenomena described in this chapter, so I won't mention it again.) Imagine a source of balls that could send a ball flying in any direction, for example a hot tungsten filament that boils off electrons. Suppose a ball begins at point P , bounces off the floor, and ends up at point Q . (Points P and Q are equally distant from the floor.
An optical realization of a quantum bouncing ball on a truncated lattice is theoretically proposed for light waves propagating near the edge of a circularly curved waveguide array. As compared to the analogous problem of matter waves bouncing on a perfectly reflecting surface under the influence of gravity, where the wave packets basically behave as classical particles, it is shown that the genuine quantum regimes of wave packet collapse and revival are accessible for the discrete optical system.
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.