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The goal is to derive the hydrogen atom energy spectrum using a geometric approach without conventional quantum mechanics.

March 18, 2026Open Access

The Hydrogen Spectrum as Torus Knot Resonance: Derivation of Atomic Energy Levels from Phase Geometry without the Schrodinger Equation

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The goal is to derive the hydrogen atom energy spectrum using a geometric approach without conventional quantum mechanics.
Develop a geometric framework based on phase oscillators on a torus.
Identify stable atomic states as torus knots categorized by (n,1).
Establish frequency and energy relationships using phase coordinates without wave functions.
Derive energy levels as E_n = -13.6/n^2 eV, matching the established hydrogen spectrum.
Achieve Balmer series wavelengths accurate to better than 0.06%.
Demonstrate that quantization occurs through the closure of phase orbits on a torus.

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This analysis derives atomic energy levels in hydrogen, revealing a geometric framework that replaces traditional quantum mechanics.

Nicolae Pascal (2026) studied this question.

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