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Electronic Transitions: A Novel Calculation Method and the End of Uncertainty

Xue Zou

Published Oct 16, 2026• 0 citations
Publisher

Abstract

Based on the open-loop and closed-loop theory of Daojing, a novel physical mechanism and calculation formula for electronic transitions are proposed. The Bohr radius and the hydrogen atom energy level formula are derived. For a total of 26 hydrogen spectral lines across the Lyman, Balmer, Paschen, Brackett, and Pfund series, the theoretical calculation results are consistent with public experimental data within four significant figures. This demonstrates that quantization is an inevitable result of closed-loop stability, rather than an artificial hypothesis, and that electronic transitions are a continuous transition of closed-loop dynamic periodic modes, rather than an instantaneous quantum jump.

DOI
10.5281/zenodo.23419160

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Based on the open-loop and closed-loop theory of Daojing, a novel physical mechanism and calculation formula for electronic transitions are proposed. The Bohr radius and the hydrogen atom energy level formula are derived. For a total of 26 hydrogen spectral lines across the Lyman, Balmer, Paschen, Brackett, and Pfund series, the theoretical calculation results are consistent with public experimental data within four significant figures. This demonstrates that quantization is an inevitable result of closed-loop stability, rather than an artificial hypothesis, and that electronic transitions are a continuous transition of closed-loop dynamic periodic modes, rather than an instantaneous quantum jump.
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This paper develops and verifies a physics-first description of the hydrogen-atom transition from the L shell (n = 2) to the K shell (n = 1) by combining the historical Bohr model of atomic structure with classical Maxwell electrodynamics. Beginning with the Coulomb force, centripetal dynamics, and Bohr’s angular-momentum quantization condition, the analysis derives the allowed orbital radii, electron velocities, angular momenta, kinetic energies, electrostatic potential energies, and total bound-state energies of the hydrogen atom. For the n = 2 → n = 1 transition, the calculated energy difference is approximately 10.2 eV, corresponding through the Planck–Einstein relation ΔE = hν to a frequency of approximately 2.47 × 10¹⁵ Hz and a wavelength near 121.6 nm, consistent with the hydrogen Lyman-α transition. The emitted radiation is then examined through Maxwell’s equations and the vacuum constitutive relations. The resulting electromagnetic-wave description yields the transverse electric and magnetic fields together with the associated electromagnetic energy density, Poynting energy flux, momentum density, radiation pressure, and Maxwell stress tensor. In this way, the atomic energy difference obtained from the Bohr model is connected quantitatively to the classical propagation properties of the outgoing electromagnetic radiation. The derivation is mathematically consistent as a historical Bohr–Maxwell reconstruction, but it does not constitute a closed dynamical theory of spontaneous emission. The essential bridge between discrete atomic energy levels and the continuous Maxwell field is supplied by the quantum frequency condition ΔE = hν. Neither classical Maxwell electrodynamics nor Newtonian orbital mechanics independently derives this relation, and classical Maxwell theory does not by itself quantize the emitted radiation energy into individual photons. A complete treatment of spontaneous emission, transition probabilities, radiative lifetimes, and photon-field quantization requires quantum mechanics and ultimately quantum electrodynamics. Accordingly, the framework is presented as a verified correspondence between established limiting descriptions rather than as a replacement for Schrödinger wave mechanics, Dirac theory, or quantum electrodynamics. By clearly separating the atomic bound-state calculation, the quantum energy-frequency bridge, and the classical electromagnetic propagation problem, the analysis provides a rigorous conceptual and mathematical foundation for subsequent Maxwell–Schrödinger, Dirac–Maxwell, quantum-electrodynamic, Maxwell–Scretching, and Scretching Quantum Chain formulations of hydrogenic radiative transitions.
We present a unified Matrix Theory (MT) framework that explains hydrogen spectral lines through two distinct but interconnected mechanisms: (1) quantized orbital transitions governed by the Qn structure of the H0 matrix, and (2) continuous-spectrum muon-mediated processes arising from high-energy electron motion along linear trajectories. The framework provides a mechanical origin for the Rydberg formula, the Lyman-α doublet, muon lifetime, and the density-dependent behavior of negative muons. All predictions are shown to agree with experimental data to within 0.03% for spectral lines and 18 ppm for muon lifetime. The model offers a deeper mechanistic understanding of phenomena traditionally described by quantum mechanics without replacing its empirical success.
The electronic transition energies from the electron clouds are the same as predicted by the Bohr model. The hydrogen atom, most probable radius of the first quantum mechanical electron cloud is identical to the radius of the first Bohr orbit. The quantum mechanical description goes much further, however. Not only does it predict a series of increasingly larger spherical electron clouds, but other electron cloud series as well. Because of their three dimensional shapes, these electron clouds sometimes are said to describe space states. The analogy to planetary orbits ends when we grasp the fact that quantum mechanical electron clouds are three-dimensional, where a Bohr orbit would be a circle lying entirely in a plane. Quantum mechanics also predicts the ionization energy of the hydrogen atom.
The existence of line spectra can be explained bymeans of the first “quantum”model of the atom, developed by Bohr in 1913. Although the Bohr model of the hydrogen atom was eventually replaced, it yields the correct values for the observed spectral lines, and gives a substantial insight into the structure of atoms in general. The following derivation has the objective of obtaining the energy levels of the Bohr atom. If we can obtain the energy levels, we can reproduce the hydrogen atom spectra. The derivation proceeds with three major elements: first, use force balance to relate the velocity to the radius of the electron orbit, then use a quantum assumption to get the radius, then solve for the energies.