Abstract This paper presents a continuous medium formulation that reproduces hydrogen spectral series with high accuracy. The model treats the hydrogen atom as standing-wave patterns in a continuous background field, providing an alternative mathematical framework for atomic bound states and spectral transitions. In addition to reproducing transition spectra with excellent precision, the formulation reveals a small but systematic deviation in absolute hydrogen energy levels that scales quadratically with the mode index n. This cumulative correction cancels identically in transition energies, explaining why it is absent from conventional spectroscopic analyses, yet is required to reproduce measured absolute term values. This contribution is invisible to transition-based spectroscopy but becomes explicit when absolute term values are examined, revealing structural information not accessible in operator-based formulations.
Abstract This paper presents a continuous medium formulation that reproduces hydrogen spectral series with high accuracy. The model treats the hydrogen atom as standing-wave patterns in a continuous background field, providing an alternative mathematical framework for atomic bound states and spectral transitions. In addition to reproducing transition spectra with excellent precision, the formulation reveals a small but systematic deviation in absolute hydrogen energy levels that scales quadratically with the mode index n. This cumulative correction cancels identically in transition energies, explaining why it is absent from conventional spectroscopic analyses, yet is required to reproduce measured absolute term values. This contribution is invisible to transition-based spectroscopy but becomes explicit when absolute term values are examined, revealing structural information not accessible in operator-based formulations.
This paper proposes a theoretical model in which the coupling between the electron and proton inside the hydrogen atom generates a confined electromagnetic field characterized by stable resonant modes. The model assumes that this field propagates at the speed of light and forms standing-wave structures confined within the atomic system. Unlike conventional interpretations that treat the electron purely as a probabilistic quantum object, the present approach considers the possibility that quantized atomic behavior emerges from resonant electromagnetic confinement.The study introduces a resonance-based framework capable of linking atomic stability to discrete electromagnetic frequencies associated with the hydrogen atom. Specific resonant frequencies and wavelength relations are derived and compared with known spectral transitions. The model further predicts the existence of internally confined electromagnetic patterns that may explain energy quantization from a wave-resonance perspective.In addition, the proposed framework offers potential reinterpretations of atomic transitions, field localization, and energy exchange mechanisms inside simple atomic systems. The work is presented as a preliminary theoretical foundation intended for further mathematical development, analytical derivation, and experimental examination.
We interpret the hydrogen atom as a binary system of coherent phase structures in which stationary states emerge from compatible geometric phase closures. We postulate a geometric quantization law for the characteristic radii of the system and show that it naturally leads to an energy spectrum possessing a finite upper bound equal to twice the Rydberg energy. The resulting energy levels can be expressed as the sum of a finite constant boundary energy and a term proportional to 1/n². The constant contribution cancels exactly in radiative transitions, yielding the Rydberg formula and reproducing the observed spectral lines of hydrogen without introducing the conventional quantum postulate for the 1/n² dependence of the energy levels. These results suggest that the hydrogen spectrum may be interpreted as the consequence of global constraints of phase coherence and topology rather than as the direct manifestation of a local dynamical law.
This paper develops an advanced unified description of the hydrogen atom wavefunction across three major theoretical levels: nonrelativistic quantum mechanics, relativistic quantum mechanics, and the Scretching–Schrödinger Equation framework. The study begins with the standard Schrödinger hydrogen wavefunction, where the electron state is represented as a complex probability amplitude labeled by the quantum numbers nnn, lll, mlm_lml, and msm_sms. It then advances to the relativistic Dirac treatment, where the scalar wavefunction is replaced by a four-component spinor labeled by nnn, κ\kappaκ, and mjm_jmj, thereby incorporating spin, relativistic energy corrections, and the natural appearance of fine-structure splitting. The central contribution of the paper is the extension of the hydrogen atom into the Scretching–Schrödinger Equation (SSE) framework, where the ordinary Schrödinger structure is preserved but constrained by a deterministic spectroscopic closure condition from the Scretching Quantum Chain (SQC) and Maxwell–Scretching Chain (MSC). In this formulation, a valid hydrogen wavefunction must not only satisfy the wave equation but must also reproduce the full physical chain ε→f→∣μ∣2→A21→Kν,\varepsilon \rightarrow f \rightarrow |\mu|^2 \rightarrow A_{21} \rightarrow K_{\nu},ε→f→∣μ∣2→A21→Kν, linking molar absorptivity, oscillator strength, transition dipole moment, Einstein spontaneous-emission coefficient, and the SQC invariant. The invariant Kν=2πe2ε0mec3=7.42165600145×10−22 sK_{\nu}=\frac{2\pi e^2}{\varepsilon_0 m_e c^3} =7.42165600145\times 10^{-22}\ \text{s}Kν=ε0mec32πe2=7.42165600145×10−22 s is derived from first principles and treated as the fixed spectroscopic time-scale anchor of the hydrogen SQC closure. The paper verifies the SQC formulation against NIST hydrogen spectral data for the Lyman series (n=2–4)(n=2–4)(n=2–4) and Balmer series (n=3–7)(n=3–7)(n=3–7), obtaining relative errors no larger than 2.7×10−42.7\times 10^{-4}2.7×10−4, with the remaining discrepancy attributed to rounding limits. It also derives the corrected radial transition integral for the 1s→2p1s\rightarrow 2p1s→2p transition, Ir0=1.2902662 a0,I_{r0}=1.2902662\,a_0,Ir0=1.2902662a0, and uses this value to support the deterministic connection between hydrogen wavefunctions, transition moments, oscillator strengths, and Einstein coefficients. The manuscript further develops the hydrogen SSE Hamiltonian in C*-algebraic form, extends the wavefunction treatment to isotope-corrected protium, deuterium, and tritium, and demonstrates that KνK_{\nu}Kν remains invariant under isotope substitution. Fine-structure effects—including Lamb-shift corrections, hyperfine splitting, and Zeeman perturbations—are examined as higher-order refinements within the SSE context rather than violations of the underlying SQC invariant. Finally, the paper maps the full SQC-to-MSC projection, showing how quantum oscillator strength projects upward into macroscopic electromagnetic observables such as absorption, refractive behavior, dielectric response, and radiative transition strength. The result is a deterministic, parameter-free quantum–electromagnetic hydrogen framework in which NRQM, RQM, SSE, SQC, and MSC are treated as connected layers of the same physical closure chain.
This paper applies the corrected Scretching–Schrödinger Equation (SSE) to the complete bound-state hydrogen atom spectrum. The ordinary Coulomb hydrogen Hamiltonian is retained as the exact benchmark operator, while the Scretching Quantum Chain (SQC) and Maxwell–Scretching Chain (MSC) are imposed as spectroscopic closure relations that connect quantum wavefunction matrix elements to measurable optical quantities, including oscillator strengths, molar absorptivity, Einstein coefficients, and transition frequencies. Using CODATA 2022 constants and the corrected hydrogen 1s→2p1s\rightarrow2p1s→2p radial transition integral, the Lyman-alpha transition is numerically verified with Kν=7.42165600145×10−22 s,K_{\nu}=7.42165600145\times10^{-22}\ \mathrm{s},Kν=7.42165600145×10−22 s, KSSE=7.50174281×10−13 m2 s,K_{\mathrm{SSE}}=7.50174281\times10^{-13}\ \mathrm{m^{2}\,s},KSSE=7.50174281×10−13 m2s, ∣μ∣SQC=3.2804 D,|\mu|_{\mathrm{SQC}}=3.2804\ \mathrm{D},∣μ∣SQC=3.2804 D, and A21=6.26169×108 s−1.A_{21}=6.26169\times10^{8}\ \mathrm{s^{-1}}.A21=6.26169×108 s−1. The corrected analytic hydrogen dipole moment is ∣μ∣analytic=3.2795 D,|\mu|_{\mathrm{analytic}}=3.2795\ \mathrm{D},∣μ∣analytic=3.2795 D, which removes the previously reported 13% discrepancy and confirms close agreement between the SQC closure value and the standard hydrogenic quantum-mechanical result. The paper then extends the corrected SSE framework across the full bound-state hydrogen spectrum by generating the Lyman, Balmer, Paschen, Brackett, Pfund, and Humphreys series from the reduced-mass Rydberg formula. In this application, the SSE does not replace the ordinary Schrödinger solution for hydrogen; instead, it constrains and verifies the standard Coulomb spectrum through spectroscopic closure. Because the hydrogen Coulomb Hamiltonian already reproduces the bound-state spectrum exactly within nonrelativistic quantum mechanics, the SQC correction term is effectively null for hydrogen, VSQC(H)≈0.V_{\mathrm{SQC}}(H)\approx 0.VSQC(H)≈0. Thus, for the hydrogen atom, the Scretching–Schrödinger Equation reduces to the ordinary Schrödinger equation while preserving the SQC and MSC spectroscopic chains as verification layers linking wavefunctions, transition moments, oscillator strengths, Einstein coefficients, and observed spectral frequencies [1–4,12].
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.
### Overview This repository and associated documentation introduce a transformative framework that shifts the traditional perspective on physical fields, relativistic quantum systems, and nuclear models. By unifying complex computational checks and precise data analysis, this work bridges theoretical innovations with practical, automated verification tools. ### Key Contributions & Scope * **Relativistic Quantum Systems:** Implements rigorous automated checks for hydrogen-like systems incorporating Dirac corrections and advanced wave equation solvers. * **Nuclear Models & Data Tables:** Features robust analytical tools for processing mass tables, binding energies, and nuclear structure parameters. * **Automated Verification Framework (`exchange_field_check_package_V3`):** Provides a comprehensive suite of modular scripts (`run_checks.sh`, modular model checkers) designed to ensure high reproducibility and precision in physical calculations. ### Contents of the Archive * `exchange_field_check_package_V3/`: Core package containing validation modules, test scripts, and historical/iterative round records. * Analytical scripts for atomic structures, nuclear properties, and comparative data verification. ### Usage & Quick Start To execute the automated verification suite, run: ```bash bash exchange_field_check_package_V3 /run_checks.sh # Summary This work presents a methodological and interpretational framework for Quantum Mechanics (QM) based on a **medium-first wave approach**. It does not introduce new physical laws or speculative forces; rather, it demonstrates how standard quantum phenomena—such as discrete spectra, wave-function collapse, and non-locality—can be naturally interpreted as **discrete exchange events** between continuous field modes and localized resonators within a wave medium. ### Key Highlights & Results - **Standard Physics Alignment**: The framework fully recovers established results of standard quantum mechanics and Quantum Electrodynamics (QED) without introducing non-standard free parameters. - **Hydrogen Atom & Dirac Spectrum**: Exact analytical and numerical match for the relativistic Hydrogen spectrum via the 2-component Dirac wave equation in a Coulomb field (reproducing the fine structure, 2P3/2 splitting, and 2S1/2−2P1/2 orbital structure). - **Helium Atom Ground State**: Verification of the two-electron configuration space representation (Hylleraas expansion) yielding E0=−2.903724 Hartree. - **Lamb Shift & Radiative Corrections**: Explains leading-order QED radiative shifts via mode-exchange dynamics with the zero-point wave field. - **Bell's Inequalities & Non-Locality**: Analyzes classical acoustic and hydrodynamic analogies (e.g., KdV solitons, walker drops), showing why 3D classical models fail Bell tests and demonstrating how non-locality is properly accommodated in ConfigurationSpace rather than standard 3D physical space. - **Thermodynamic & Biological Scales**: Connects medium viscosity, field density, and discrete thermal exchange equations (kBT) across scales ranging from chemical bonding to biological systems. ### Scope & Intent This work serves as an **educational, interpretational, and computational bridge** to demystify standard quantum mechanics. It provides Python/SciPy verification scripts to allow independent reproduction of all wave-equation solutions, spectral distributions, and mode-coupling c alculations. A single two-term law, dθ/dt = A − ρθ, with a toll m ≥ 1 at every door, tested against measured data in 511 rounds with the pass mark written before each number. 33 claims audited: 12 derived, 3 computed, 3 agree with measurement, 1 prediction (carrier 11 869.5 Hz, never measured), 4 parameters, 6 open, 4 killed. All failures published. 28 checks re-run from the package.