Deriving the Schrödinger Equation from Source-Free Maxwell Dynamics
Preferred Frame Writing — January 2026
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One-Sentence Summary
We derive the Schrödinger equation and the emergence of Planck's constant as the narrow-band limit of classical Maxwell wave dynamics on a toroidal standing mode.
Summary
Maxwell's equations for electromagnetism in source-free vacuum predict discrete energies when an electromagnetic field forms a self-confined toroidal standing pattern. For any component $F(\mathbf{r},t)$ of the electromagnetic fields $\mathbf{E}, \mathbf{B}$, we isolate the forward-time spectral part, keep all derivative terms exactly, and obtain —within a rigorously bounded, bandwidth-squared remainder— the Schrödinger equation. Planck’s constant and the inertial mass thus emerge not as fundamental constants, but as geometric properties of the fundamental toroidal mode ($E_{11}, \omega_{11}$).
Keywords
Keywords: Maxwell Equations, Toroidal Quantization, Analytic Signal, Emergent Quantum Mechanics, Rydberg Ladder