## Version 2 - Corrected and Updated This is a corrected and updated version of the Scretching-Schrödinger Equation (SSE) framework. Version 2 incorporates the following improvements based on NIST Atomic Spectra Database (ASD) 2024 data: ### Key Corrections:- **Hydrogen**: Updated to f = 0.4162 (1s→2p Lyman-α multiplet) with correct degeneracy g₁/g₂ = 2/6- **Lithium**: Updated to full multiplet oscillator strength f = 0.7416 (2s→2p)- **Beryllium**: Updated to f = 1.340 (NIST ASD)- **Units**: Transition dipole moments |μ|² now correctly reported in C²m²- **K_SSE**: Corrected to 7.502×10⁻¹³ m²·s- **N_valence**: Corrected formula: N_val = (4.32×10⁻⁹)·Σ(ε_i·Δν̄_i) ### Results:The framework now achieves:- H (Lyman-α): 0.05% agreement with NIST A₂₁- He: 0.02% agreement- Li: 0.67% agreement- Be: 9.9% (within NIST uncertainty rating C, ±25%)- B: 0.08% agreement- C: 12.2% (consistent with multi-configuration complexity) ### AbstractBackground: One hundred years after Schrödinger's wave equation, a fundamental disconnect persists between quantum mechanical predictions and experimental spectroscopy. Wavefunctions are calculated first, then compared to spectra, with no mechanism for experimental data to constrain the solutions. Objective: We present the corrected Scratching-Schrödinger Equation (SSE), a reformulation that incorporates experimental UV-Vis absorption data as deterministic constraints within the wave equation. Methods: Starting from the Scratching Quantum Chain (SQC) relations linking oscillator strength f to transition dipole moment |μ|², and the Maxwell-Scratching Chain (MSC) connecting molar absorptivity ε to f, we derive a constrained Schrödinger equation with a V_SQC term enforcing exact agreement with experimental transition dipoles. Results: Applied to the first six elements (H, He, Li, Be, B, C) with corrected NIST ASD input data, the SSE reproduces experimental transition probabilities with high precision and identifies sources of uncertainty in complex cases. Conclusion: A century after Schrödinger's breakthrough, the SSE framework establishes a deterministic connection between wave mechanics and experimental spectroscopy. ### Related Identifiers- **Previous version**: 10.5281/zenodo.18805791
Version 2 - Corrected and Updated This is a corrected and updated version of the Scretching-Schrödinger Equation (SSE) framework. Version 2 incorporates the following improvements based on NIST Atomic Spectra Database (ASD) 2024 data: ### Key Corrections:- Hydrogen: Updated to f = 0.4162 (1s→2p Lyman-α multiplet) with correct degeneracy g₁/g₂ = 2/6- Lithium: Updated to full multiplet oscillator strength f = 0.7416 (2s→2p)- Beryllium: Updated to f = 1.340 (NIST ASD)- Units: Transition dipole moments |μ|² now correctly reported in C²m²- K_SSE: Corrected to 7.502×10⁻¹³ m²·s- N_valence: Corrected formula: N_val = (4.32×10⁻⁹)·Σ(ε_i·Δν̄_i) ### Results:The framework now achieves:- H (Lyman-α): 0.05% agreement with NIST A₂₁- He: 0.02% agreement- Li: 0.67% agreement- Be: 9.9% (within NIST uncertainty rating C, ±25%)- B: 0.08% agreement- C: 12.2% (consistent with multi-configuration complexity) ### AbstractBackground: One hundred years after Schrödinger's wave equation, a fundamental disconnect persists between quantum mechanical predictions and experimental spectroscopy. Wavefunctions are calculated first, then compared to spectra, with no mechanism for experimental data to constrain the solutions. Objective: We present the corrected Scratching-Schrödinger Equation (SSE), a reformulation that incorporates experimental UV-Vis absorption data as deterministic constraints within the wave equation. Methods: Starting from the Scratching Quantum Chain (SQC) relations linking oscillator strength f to transition dipole moment |μ|², and the Maxwell-Scratching Chain (MSC) connecting molar absorptivity ε to f, we derive a constrained Schrödinger equation with a V_SQC term enforcing exact agreement with experimental transition dipoles. Results: Applied to the first six elements (H, He, Li, Be, B, C) with corrected NIST ASD input data, the SSE reproduces experimental transition probabilities with high precision and identifies sources of uncertainty in complex cases. Conclusion: A century after Schrödinger's breakthrough, the SSE framework establishes a deterministic connection between wave mechanics and experimental spectroscopy. ### Related Identifiers- Previous version: 10.5281/zenodo.18805791
## Version 2 - Corrected and Updated This is a corrected and updated version of the Scretching-Schrödinger Equation (SSE) framework. Version 2 incorporates the following improvements based on NIST Atomic Spectra Database (ASD) 2024 data: ### Key Corrections:- **Hydrogen**: Updated to f = 0.4162 (1s→2p Lyman-α multiplet) with correct degeneracy g₁/g₂ = 2/6- **Lithium**: Updated to full multiplet oscillator strength f = 0.7416 (2s→2p)- **Beryllium**: Updated to f = 1.340 (NIST ASD)- **Units**: Transition dipole moments |μ|² now correctly reported in C²m²- **K_SSE**: Corrected to 7.502×10⁻¹³ m²·s- **N_valence**: Corrected formula: N_val = (4.32×10⁻⁹)·Σ(ε_i·Δν̄_i) ### Results:The framework now achieves:- H (Lyman-α): 0.05% agreement with NIST A₂₁- He: 0.02% agreement- Li: 0.67% agreement- Be: 9.9% (within NIST uncertainty rating C, ±25%)- B: 0.08% agreement- C: 12.2% (consistent with multi-configuration complexity) ### AbstractBackground: One hundred years after Schrödinger's wave equation, a fundamental disconnect persists between quantum mechanical predictions and experimental spectroscopy. Wavefunctions are calculated first, then compared to spectra, with no mechanism for experimental data to constrain the solutions. Objective: We present the corrected Scratching-Schrödinger Equation (SSE), a reformulation that incorporates experimental UV-Vis absorption data as deterministic constraints within the wave equation. Methods: Starting from the Scratching Quantum Chain (SQC) relations linking oscillator strength f to transition dipole moment |μ|², and the Maxwell-Scratching Chain (MSC) connecting molar absorptivity ε to f, we derive a constrained Schrödinger equation with a V_SQC term enforcing exact agreement with experimental transition dipoles. Results: Applied to the first six elements (H, He, Li, Be, B, C) with corrected NIST ASD input data, the SSE reproduces experimental transition probabilities with high precision and identifies sources of uncertainty in complex cases. Conclusion: A century after Schrödinger's breakthrough, the SSE framework establishes a deterministic connection between wave mechanics and experimental spectroscopy. ### Related Identifiers- **Previous version**: 10.5281/zenodo.18805791
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].
This paper derives the Scretching–Schrödinger Equation (SSE) from first principles by unifying the standard probability-amplitude foundation of quantum mechanics with experimentally anchored spectroscopic closure. Beginning with the Born wavefunction Ψ(r,t), the Planck–de Broglie relations E=ℏωE=\hbar\omegaE=ℏω and p=ℏkp=\hbar kp=ℏk, probability conservation, and the requirement that Hamiltonian evolution be self-adjoint, the paper reconstructs the ordinary Schrödinger equation as the baseline unitary wave equation for real-energy quantum systems. The derivation then extends the ordinary Hamiltonian by imposing the Scretching Quantum Chain (SQC) as a transition-level constraint connecting wavefunction matrix elements, transition dipole moments, oscillator strengths, Einstein radiative coefficients, molar absorptivity, and Maxwell–Scretching optical closure. In this formulation, the Scretching–Schrödinger Hamiltonian is not merely a phenomenological energy shift but a constrained operator equation, HSSE=H0+VSQC,H_{\mathrm{SSE}} = H_0 + V_{\mathrm{SQC}},HSSE=H0+VSQC, with the operator domain D(HSSE)=D(H0)∩D(VSQC).D(H_{\mathrm{SSE}})=D(H_0)\cap D(V_{\mathrm{SQC}}).D(HSSE)=D(H0)∩D(VSQC). Here, VSQCV_{\mathrm{SQC}}VSQC is treated as a symmetric or self-adjoint spectroscopic correction operator that preserves the mathematical structure required for physical quantum evolution while enforcing experimentally measurable spectroscopic constraints. Hydrogen is used as the zero-correction benchmark because its Coulomb Hamiltonian already provides an exact analytic reference for bound-state spectra and transition moments. The corrected 1s→2p1s\rightarrow 2p1s→2p analytic dipole agrees with the SQC-derived dipole to approximately 0.03%, supporting the interpretation that the SSE reduces to ordinary Schrödinger theory when no additional spectroscopic correction is required. The resulting framework therefore presents the SSE as a rigorously constrained extension of quantum mechanics: ordinary wave mechanics supplies the self-adjoint dynamical core, while the SQC and Maxwell–Scretching chains impose measurable optical, radiative, and molecular-spectroscopic closure conditions on allowed transitions.
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.
Introduction The Scretching–Schrödinger Equation (SSE) is developed as a quantum-mechanical framework that imposes bidirectional spectroscopic closure between the wavefunction and the measured spectrum, ψ⟷spectrum,\psi \longleftrightarrow \mathrm{spectrum},ψ⟷spectrum, through the Scretching Quantum Chain (SQC) invariant, Kν=7.42165600145×10−22 s.K_{\nu}=7.42165600145\times10^{-22}\ \mathrm{s}.Kν=7.42165600145×10−22 s. In its original non-relativistic form, the SSE modifies the ordinary Schrödinger framework by requiring that the wavefunction not only solve an eigenvalue problem, but also satisfy a deterministic spectroscopic closure condition. This condition links transition matrix elements, oscillator strengths, Einstein coefficients, absorption parameters, and measured transition frequencies. The result is a constrained quantum theory in which ψ\psiψ generates spectra, while experimental spectra feed back as admissibility conditions on ψ\psiψ. This paper systematically recasts the SSE across the full hierarchy of modern quantum theory: ordinary quantum mechanics, relativistic quantum mechanics, quantum electrodynamics, electroweak theory, and quantum chromodynamics. The objective is to show that the SSE closure principle can be expressed not merely as a correction to non-relativistic wave mechanics, but as a generalized spectroscopic constraint compatible with the major operator, field, and gauge-theoretic structures of twentieth- and twenty-first-century physics. The first stage reformulates the SSE within ordinary quantum mechanics in both the Schrödinger wave-mechanical formalism and the Heisenberg matrix-mechanical formalism. In the Schrödinger picture, the SSE is written as a constrained wave equation, iℏ∂ψ∂t=(H^0+V^SQC)ψ,i\hbar\frac{\partial \psi}{\partial t} = \left(\hat{H}_0+\hat{V}_{\mathrm{SQC}}\right)\psi,iℏ∂t∂ψ=(H^0+V^SQC)ψ, where H^0\hat{H}_0H^0 is the ordinary Hamiltonian and V^SQC\hat{V}_{\mathrm{SQC}}V^SQC is the spectroscopic closure operator. In the Heisenberg picture, the same closure is recast into operator evolution and matrix-element constraints, showing that the SSE condition is independent of representation. The wavefunction, operator algebra, transition amplitude, and spectrum therefore become mutually constrained expressions of the same physical closure. The second stage extends the SSE into relativistic quantum mechanics (RQM) through the Klein–Gordon and Dirac equations. The Klein–Gordon–SSE form imposes SQC closure on relativistic scalar fields, while the Dirac–SSE form imposes the same closure on four-component spinor states. These extensions preserve relativistic covariance, introduce proper mass-shell structure, and connect the SSE to spin, antimatter, fine structure, and relativistic transition amplitudes. In this form, the SSE becomes a spectroscopic constraint on relativistic wave equations rather than a purely non-relativistic correction. The third stage recasts the framework within quantum electrodynamics (QED). Here the SSE closure principle is applied to photon exchange, electron self-energy, vertex corrections, Ward identity structure, and the running electromagnetic coupling. The QED–SSE formulation treats the SQC closure functional as a constraint on radiative processes, linking Feynman amplitudes to spectroscopic observables. The photon propagator, electron propagator, interaction vertex, and renormalized coupling are therefore interpreted not only as field-theoretic objects, but also as carriers of measurable spectroscopic closure. The fourth stage extends the formalism into the Glashow–Salam–Weinberg (GSW) electroweak theory, whose gauge group is SU(2)L×U(1)Y.SU(2)_L\times U(1)_Y.SU(2)L×U(1)Y. In this setting, the SSE closure principle is applied to weak and electromagnetic transitions after spontaneous symmetry breaking. The framework is developed through the electroweak gauge fields, Higgs mechanism, weak mixing angle, and the resulting W±W^{\pm}W±, Z0Z^0Z0, and photon fields. The masses of the weak bosons and the structure of charged and neutral currents are incorporated into an SSE weak-transition closure law, allowing the SQC framework to be tested against electroweak-scale processes. The fifth stage applies the SSE to quantum chromodynamics (QCD), based on the color gauge group SU(3)c.SU(3)_c.SU(3)c. Because QCD includes gluon self-interaction, asymptotic freedom, confinement, and non-Abelian color dynamics, it provides the most demanding field-theoretic extension of the SSE. The QCD–SSE formulation introduces a color-closure functional designed to connect quark and gluon transition amplitudes with spectroscopic or hadronic observables. In this context, the SSE is not presented as a replacement for QCD, but as a proposed closure constraint that may organize measurable color-sector transitions, confinement-scale behavior, and hadronic spectra. For each theoretical level, this paper derives a corresponding SSE-extended equation, lists the relevant Feynman rules, presents representative Feynman diagrams, and provides numerical examples with verified solutions. The analysis traces the mathematical progression from wave mechanics to matrix mechanics, from non-relativistic quantum theory to relativistic wave equations, from single-particle dynamics to QED radiative corrections, from Abelian gauge theory to electroweak unification, and finally from electroweak theory to the non-Abelian color structure of QCD. The primary textbook sources abstracted for the gauge-theory and particle-physics development are Aitchison and Hey’s Gauge Theories in Particle Physics, Volumes 1 and 2, including the third and fourth edition treatments of quantum electrodynamics, electroweak theory, and quantum chromodynamics [1,2]. All numerical constants used in the worked examples are taken from CODATA 2022, ensuring that the calculations are anchored to contemporary reference values. The central claim advanced in this paper is that the SSE may be written as a representation-independent and scale-extendable spectroscopic closure principle. In ordinary quantum mechanics, it constrains wavefunctions and operator matrix elements. In RQM, it constrains relativistic scalar and spinor amplitudes. In QED, it constrains radiative amplitudes and renormalized electromagnetic interactions. In GSW theory, it constrains weak and electroweak transitions. In QCD, it proposes a color-sector closure functional for non-Abelian gauge dynamics. Thus, the SSE is developed as a candidate bridge between measured spectra, quantum amplitudes, and gauge-field theory across the full hierarchy of modern quantum physics.
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.