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.