What question did this study set out to answer?

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

Key Points

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.

Abstract

We derive the hydrogen atom energy spectrum Eₙ = -13. 6/n² eV from a purely geometric framework in which the electron is a phase oscillator on a torus. Stable atomic states correspond to (n, 1) torus knots: closed orbits where the radial phase completes exactly n cycles per angular orbit. The frequency hierarchy omegaₙ = omega₁/n³ follows from Kepler's law in phase coordinates. The phase velocity vₙ = v₁/n and energy Eₙ proportional to vₙ² = 1/n² follow algebraically. The energy scale E₁ = (1/2) alpha² mₑ c² = 13. 6 eV is set by the fine structure constant alpha = 1/137, interpreted as the electromagnetic phase coupling strength. The Balmer series wavelengths match measurements to better than 0. 06%. No wave functions, no Coulomb potential, and no Schrodinger equation are used. Atomic quantization emerges from the requirement that phase orbits close on a torus.

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

Key Points

Abstract

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