Emergence VI: Atomic and Optical Physics from Wave Intersections on a Pre-Geometric Canvas

This paper derives the fundamental laws of atomic and optical physics from the canvas model, a pre-geometric framework in which all of physics emerges from wave intersections on a primordial canvas. The derivation begins with the six core equations of the canvas model and proceeds step by step to the standard results of atomic and optical physics. The Lamb shift, a tiny energy difference between the two otherwise degenerate 2S and 2P states of hydrogen that played a crucial role in the development of quantum electrodynamics, is derived from the zero-point oscillations of the canvas vacuum. The calculation follows Bethe's classic method, but here the vacuum fluctuations are not a mathematical artifact of quantum field theory—they are real oscillations of the canvas fields. The result agrees with the experimentally measured shift of approximately 1057 megahertz. Hyperfine splitting, the energy difference between spin-aligned and spin-opposite configurations of the electron and nucleus in hydrogen, is derived from the Fermi contact interaction of closed wave spins. The famous twenty-one centimeter line emerges naturally with its precisely measured frequency of 1420 megahertz, giving the model an exact numerical prediction that matches observation. The Zeeman effect, the splitting of atomic energy levels in a magnetic field, is derived with the full Landé g-factor, showing how orbital and spin angular momenta combine in the canvas geometry. The Stark effect, the shifting and splitting of atomic levels in an electric field, is derived for both the linear case in hydrogen and the quadratic case in non-degenerate atoms. The Einstein A and B coefficients, which govern spontaneous emission, absorption, and stimulated emission, are derived from threshold crossing probabilities. Spontaneous emission is shown to be a threshold crossing triggered by canvas vacuum fluctuations, not an unexplained axiom of quantum theory. The relation between absorption and stimulated emission follows from the symmetry of the canvas wave equation. Rabi oscillations, the periodic exchange of energy between a two-level atom and a resonant driving field, are derived from the canvas wave equation in the rotating wave approximation. The oscillation frequency is given by the dipole matrix element divided by Planck's constant—the standard result, but now emerging from coherent threshold crossings between the atom and the canvas field. The Sagnac effect, a phase shift in a rotating interferometer proportional to the enclosed area, is derived from the canvas wave equation in a rotating frame. The rotation introduces a Coriolis-like term that breaks the symmetry between co-rotating and counter-rotating waves, giving a purely geometric phase shift that depends only on the area enclosed, not on the shape of the path. The Faraday effect, the rotation of polarized light passing through a medium in a magnetic field, is derived from magnetic circular birefringence. The magnetic field couples to the closed wave spins in the medium, creating different refractive indices for left and right circular polarizations. The rotation angle is proportional to the magnetic field strength and the path length, with the constant of proportionality given by the Verdet constant. Optical coherence, the persistence of phase relationships between canvas waves, is derived using the Wiener-Khinchin theorem. The coherence time is inversely proportional to the linewidth of the source, and the coherence length is the speed of light times the coherence time. Decoherence occurs when interactions with environmental canvas fields randomize the phases. All derivations follow from the six core equations of the canvas model and reproduce the standard results of atomic and optical physics. The canvas model provides physical mechanisms for phenomena that standard quantum theory postulates without explanation. This paper is part of the Emergence series, which collectively derives all of physics from wave intersections on a pre-geometric canvas. Further derivations of quantum foundations, black hole physics, cosmology, particle physics, condensed matter, nuclear physics, and classical physics are presented in the companion Emergence papers.