This technical report presents the current status of the Rasan Atomic Model (RAM-0.2), a nonlinear phenomenological extension of the non-relativistic Schrödinger equation for the hydrogen atom. Key contents:- Self-consistent numerical solution of the radial equation for the 1s and 2s states- Proposed (but incomplete) multi-particle and radiation-field generalization- Clear distinction between the energy-functional gap and the stationary nonlinear eigenvalue gap- Conditional spectroscopic sensitivity estimates for the 1S–2S transition- Explicit statement that the model is still a research hypothesis, not a complete QED theory, and that no unique spectral prediction is yet available The accompanying files include:- Reproducible Python code for the radial SCF solver- CSV tables of grid convergence and spectroscopic comparison- Sensitivity plot Scientific status: Research hypothesis under development. Not a confirmed theory, not a violation of standard quantum mechanics, and not yet suitable for patent claims or claims of being a complete atomic model.
This technical report presents the current status of the Rasan Atomic Model (RAM-0.2), a nonlinear phenomenological extension of the non-relativistic Schrödinger equation for the hydrogen atom. Key contents:- Self-consistent numerical solution of the radial equation for the 1s and 2s states- Proposed (but incomplete) multi-particle and radiation-field generalization- Clear distinction between the energy-functional gap and the stationary nonlinear eigenvalue gap- Conditional spectroscopic sensitivity estimates for the 1S–2S transition- Explicit statement that the model is still a research hypothesis, not a complete QED theory, and that no unique spectral prediction is yet available The accompanying files include:- Reproducible Python code for the radial SCF solver- CSV tables of grid convergence and spectroscopic comparison- Sensitivity plot Scientific status: Research hypothesis under development. Not a confirmed theory, not a violation of standard quantum mechanics, and not yet suitable for patent claims or claims of being a complete atomic model.
### 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.
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 presents a comprehensive reformulation and extension of the Scretching–Schrödinger Equation (SSE) across three complementary quantum-mechanical formalisms. First, the SSE is developed in its wave-mechanical operator form by connecting the spectroscopically constrained Hamiltonian structure to the Tao–Hamilton operator framework and the Zhang–Svidzinsky probability-amplitude derivation of the Schrödinger equation. Second, the SSE is recast in Heisenberg matrix-mechanical form, where its constraints appear as operator and matrix equations governed by explicit spectroscopic closure conditions. Third, rigorous relativistic extensions are constructed through the Klein–Gordon–SSE and Dirac–SSE equations, including spinor structure, gauge covariance, and compatibility with standard relativistic quantum mechanics. The paper further integrates hyperfine-structure, Zeeman-effect, and Lamb-shift literature; the Andrews pure-wave-system derivation; and Kocher et al.’s nonlinear Schrödinger numerical methods to show that the SSE closure operator, V^SQC\hat{V}_{\mathrm{SQC}}V^SQC, can be expressed within every major formalism of quantum theory. New derived results include the SSE Heisenberg equation of motion, SSE density matrix, SSE path-integral action, SSE propagator, and a dimensionally correct SI optical susceptibility equation. All numerical results are triple-verified using CODATA 2022 constants and benchmarked against NIST reference standards where applicable. Hydrogen is used throughout as the zero-correction benchmark, with V^SQC(H)≈0\hat{V}_{\mathrm{SQC}}(\mathrm{H}) \approx 0V^SQC(H)≈0 confirmed by the Lyman-α\alphaα dipole comparison, giving δμ=0.027%\delta_{\mu} = 0.027\%δμ=0.027%. This establishes hydrogen as the reference case in which the ordinary Schrödinger, Heisenberg, Klein–Gordon, and Dirac descriptions remain fully consistent with the SSE spectroscopic closure, while also providing the foundation for applying SSE corrections to more complex atoms, molecules, and quantum systems.
The purely algebraic technique associated with the creation and annihilation operators to resolve the radial equation of Hydrogen-like atoms (HLA) for generating the bound energy spectrum and the corresponding wave functions is suitable for many calculations in quantum physics. However, the analytical approach with series is extensively used based on wave mechanics theory in most of quantum textbooks. Indeed, much More complete than the old solution of Schrödinger's time-independent differential equation (TISE), one can simply earn all quantum information of a system by using the operational method. In addition to earlier two models, including the quantum harmonic oscillator and the total angular momentum, it can undoubtedly be a third fundamental model to solve Schrödinger's eigenvalue equation of the HLA systems in Hilbert space similar to the harmonic oscillator. We will illustrate how systematically making an appropriate groundwork to discover the coherent states can lead to providing the energy quantization and normalized radial wave functions attached to the matrix representation without additional assumptions.
This paper presents a rigorous first-principles derivation of a master equation connecting the full Standard Model of particle physics to the nonrelativistic quantum mechanics of the hydrogen atom. The derivation begins with the quantum chromodynamic and electroweak structure of the Standard Model, including QCD with quarks and gluons and the electroweak symmetry group SU(2)ₗ × U(1)ᵧ. Through renormalization-group reduction, effective-field-theory matching, and the Feynman–Vernon influence-functional formalism, high-energy degrees of freedom are systematically integrated out to obtain a reduced open-system description of atomic physics. The resulting reduced dynamics take the form of a Nakajima–Zwanzig master equation. In the Markovian limit, this equation reduces to a Lindblad master equation whose effective jump operators describe the major physical transitions along the QCD → QED → QM chain: confinement of quarks into the proton, radiative emission and absorption through Einstein A and B coefficients, and weak neutral-current corrections. The framework therefore treats atomic spectroscopy not as an isolated nonrelativistic model, but as the low-energy, open-quantum-system limit of the Standard Model. Within this construction, the deterministic sector of the master equation reproduces the Scretching Quantum Chain (SQC) and the Maxwell–Scretching Chain (MSC). The central invariant recovered is Kᵥ = 2πe²/(ε₀mₑc³) = 7.42165600145 × 10⁻²² s, which is verified against NIST hydrogen Lyman transitions for n = 2–4 and Balmer transitions for n = 3–7, with relative errors at or below 2.7 × 10⁻⁴. The analysis also shows that the Schrödinger, Bohr, and Sommerfeld descriptions arise as successive approximations of the same master-equation hierarchy, rather than as disconnected models. In this setting, the corrected hydrogen radial transition integral is derived as Iᵣ₀ = 1.2902662 a₀ for the 1s → 2p transition. The paper’s main contribution is a unified deterministic bridge from quantum field theory to atomic spectroscopy. It connects QCD confinement, QED radiative structure, Lindblad open-system dynamics, and nonrelativistic hydrogen wavefunctions within a single mathematical chain. The resulting framework supports applications beyond hydrogen, including UV photon–DNA interaction models, Feynman–Scretching molecular-QED diagrams, Maxwell–Scretching optical closure, and the C*-algebraic formulation of the hydrogen wavefunction. In this way, the master equation provides a mathematically structured pathway from the Standard Model to molecular and biological spectroscopy.
This paper discusses the relativistic Schrödinger differential wave equation for the hydrogen atom and hydrogen-like cations, which have measured ionization energies from hydrogen to the Cu 28+ hydrogen-like cation.The analytical solution to the relativistic Schrödinger differential wave equation correlates with experimental data if the angular momentum quantum number l >0.For quantum levels with l = 0 , the analytical solutions predict ionization energy values larger than that measured for the hydrogen atom and hydrogen-like cations.Also, beyond the element erbium, Z > 68 for l=0 quantum states, the analytical solution yields energy values that are complex numbers.This is similar for the Dirac relativistic differential wave equation which fails for nuclei that have an atomic number larger than 137.Using measured 1s ionization energy values of hydrogen-like cations, a numerical solution to the relativistic Schrödinger differential wave equation is possible since an analytical one is not when the measured ionization energy is different from the analytical result.In addition, since the analytical solutions to the radial part of the relativistic Schrödinger differential wave equation are complicated, normalized numerical solutions are less difficult to obtain using Microsoft Excel.Most differential equations do not have analytical solutions, and, therefore, only numerical solutions are possible utilizing boundary or initial conditions. Classical Schrödinger Differential Wave Equation and Wave-Particle Duality in Quantum Mechanics