π in Fundamental Quantum Systems - (:unkn) unknown (Publisher, 2025)
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.
This dataset accompanies the scientific article exploring the fundamental role of π in quantum mechanics. It demonstrates how π naturally arises in three cornerstone systems: the quantum harmonic oscillator, the hydrogen atom, and the Fourier transform linking position and momentum. The figures and visualizations included illustrate: Gaussian wavefunction normalization in the harmonic oscillator 3D probability density and spherical harmonics of the hydrogen atom Fourier duality showing the conservation of total probability These results highlight π as a universal invariant in quantum mechanics, regulating normalization, symmetry, and coherence. Far from being a geometric relic, π emerges as the mathematical signature of balance between discrete quantization and continuous probability.