This paper develops the quantum dynamical layer of the Quantum Lattice Model (QLM) by showing how the standard wave equations of quantum mechanics arise as continuum descriptions of coherent phase-action transport on a discrete Planck-scale lattice. The framework is constructed from the per-radian Planck primitives {ℏ, ℓP, tP}, where each lattice tick advances physical phase by one radian and transports one quantum of reduced action. The corresponding phase-action and phase-flow relations are dS = ℏ dθ and E = ℏ dθ/dτ, while the invariant lattice transport identity c = ℓP/tP fixes the maximal causal phase-transport speed of the lattice. Starting from symmetric nearest-neighbor phase transport between Planck cells, the continuum limit yields the relativistic wave operator (1/c²)∂t² − ∇². Introducing intrinsic rest-phase rotation with frequency ω0 = mc²/ℏ produces the Klein-Gordon equation as the continuum description of synchronized spatial phase transport and local rest-phase rotation. A first-order relativistic factorization yields the Dirac equation, while extraction of the rapidly rotating rest-phase carrier produces the nonrelativistic Schrödinger equation governing slow envelope dynamics. Within this framework, the quantum wavefunction is interpreted as a coherent phase-envelope amplitude describing phase-action transport across the lattice. The conserved density ρ = |φ|² arises from the continuity equation of Schrödinger evolution, and the associated current takes the phase-flow form J = (ℏ/m)ρ∇θ. The paper also shows how interaction sectors enter the same transport framework as modular coupling layers. Scalar potentials contribute additional local phase rotation to the slow envelope, while routing-weighted transport modifies the effective capacity of lattice links. Coulomb coupling and weak-field gravitational coupling are presented as examples: the Coulomb potential appears as a scalar phase-rotation term, and gravitational routing modulation of the rest-phase carrier yields the weak-field potential V = mΦ. These results organize the Schrödinger, Klein-Gordon, and Dirac equations as successive continuum limits of deterministic phase-action transport on the Planck lattice, providing the quantum dynamical layer of the Quantum Lattice Model.
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.
A new relativistic quantum wave equation has been derived by applying the quantum prescription to the momentum and the kinetic energy rather than to the momentum and the total energy, since after all it is the kinetic energy that generates the momentum. The resulting equation reduces to the Schrodinger equation in the nonrelativistic limit and to the Klein-Gordon equation for “massless particles” in the relativistic limit, i.e., if the velocity of the particle approaches that of light, c. For massive particles in general, the new equation deviates from the Klein-Gordon equation. The same equation is shown to decouple according to the Dirac formalism, yielding a modified form of Dirac equation. When applied to a rest particle, the modified Dirac equation is shown to avoid a negative energy solution and instead include a constant solution. The other, the time-dependent particle solution of the modified Dirac equation, has the characteristic frequency Mc2/(ћ/2) , i.e., twice those of the Dirac solutions, Mc2/ћ.
The initial purpose is to add two physical origins for the outstandingly clear mathematical description that Dirac has left in his Principles of Quantum Mechanics. The first is the “internal motion” in the wave function of the electron that is now expressed through dispersion dynamics; the second is the physical origin for mathematical quantization. Bohr’s model for the hydrogen atom was “the greatest single step in the development of the theory of atomic structure.” It leads to the Schrodinger equation which is non-relativistic, but which conveniently equates together momentum and electrostatic potential in a representation containing mixed powers. Firstly, we show how the equation is expansible to approximate relativistic form by applying solutions for the dilation of time in special relativity, and for the contraction of space. The adaptation is to invariant “harmonic events” that are digitally quantized. Secondly, the internal motion of the electron is described by a stable wave packet that implies wave-particle duality. The duality includes uncertainty that is precisely described with some variance from Heisenberg’s axiomatic limit. Harmonic orbital wave functions are self-constructive. This is the physical origin of quantization.
### 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.
The Schrödinger equation is not a postulate. It is the non-relativistic limitof the Klein-Gordon equation that governs Hopf winding modes in the Jobsoncell medium. The derivation chain has zero free parameters: 1. Jobson medium: K = 1/eps_0, rho = mu_0, c = sqrt(K/rho) [PROVEN] 2. Free pressure wave: omega = c*k (massless) [QM1] 3. Hopf winding rest energy = m_p*c^2 -> Compton frequency omega_C = m_p*c^2/hbar [QM2; the Compton wavelength IS lambda_p = hbar_c/m_p = Zone 1 boundary] 4. Winding dispersion: omega^2 = c^2*k^2 + omega_C^2 [Klein-Gordon, QM3] 5. Non-relativistic limit (v << c): omega_KG -> omega_C + hbar*k^2/(2*m_p) Remove rest-mass oscillation phi = psi * exp(-i*m_p*c^2*t/hbar): i*hbar * dpsi/dt = -(hbar^2/2*m_p) * nabla^2 * psi [SCHRODINGER, QM5] 6. Born rule: P(x) = |psi(x)|^2 from medium wave energy density [QM6] The Schrödinger equation is quantum mechanics. It is now derived from thesame medium (Jobson cell lattice) that gives alpha, G, the Higgs mass, andquantum entanglement. No additional postulates or free parameters. Consequences: - Double-slit: Hopf winding fields span both slits simultaneously. Interference = constructive/destructive Huygens superposition in medium. - Which-path: Zone 3 field decoherence resolves winding at r < r_lock(T). r_lock(300K) = 340 fm; coherence preserved for macroscopic slit separations. - Delayed choice: no paradox. Medium configuration never had a single-slit address. "Measurement" = local perturbation resolving global winding. Timing is irrelevant; topology is not retroactively changed. NEW PREDICTIONS (not in standard QM): - Minimum electron slit width for coherent diffraction: 2*lambda_bar_e = 772 fm (below this, slit-edge Zone 3 fields decohere the electron winding) - Minimum proton slit width: 2*lambda_bar_p = 0.421 fm = r_grind (nuclear hard core) - Fringe visibility vs temperature: V(T) ~ exp(-r_sep / r_lock(T)) where r_lock(T) = (alpha*hbar_c*r_p^2 / k_B*T)^(1/3)