Welcome to AfricArXiv

This initiative showcases UbuntuNet's commitment to fostering knowledge sharing, collaboration, and accessibility within the African research community. With AfricArxiv, researchers across the continent have a dedicated platform to disseminate their findings, making them accessible to a global audience. By facilitating open access to scholarly work, UbuntuNet Alliance plays a pivotal role in advancing the principles of open science, enhancing research visibility, and driving innovation across Africa.

 

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π in Fundamental Quantum Systems
(Publisher, 2025-11-05)
In this article, I explore the profound and recurrent emergence of the mathematical constant π within the foundations of quantum mechanics. Through detailed examination of three cornerstone systems—the quantum harmonic oscillator, the hydrogen atom, and the Fourier duality linking wave and particle domains—I demonstrate that π is not merely a geometrical artifact, but a structural necessity for the internal consistency of quantum theory. In the harmonic oscillator, π arises through Gaussian normalization, ensuring that the total probability of a quantum state remains unity. In the hydrogen atom, π governs the spherical harmonics that describe the spatial symmetry of atomic orbitals, embedding π directly into the fabric of atomic structure. In Fourier transformations, π regulates the conversion between conjugate variables, such as position and momentum, preserving the unitarity and coherence of quantum information. These manifestations point toward a unifying interpretation: π acts as the mathematical signature of equilibrium between the discrete and the continuous, between quantization and continuity, probability and certainty. Far from being an incidental constant, π represents the invisible regulator of quantum harmony—the silent measure that maintains coherence and symmetry throughout the quantum universe.
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