Parwani and Tabia established in 2006 that in information-theoretic nonlinear extensions of the Schrödinger equation the energy shift of an excited state is carried predominantly by the nodes of the unperturbed wavefunction, while nodeless ground states receive only subleading shifts. This paper shows that for the rotation-invariant branch of that family, regulated by the small components of the Dirac spinor as proposed by Liew and Parwani, the node contribution closes in a single expression: each node of the density contributes exactly −4πC²/ε to the shift integral, where C is the slope of the unnormalised wavefunction at the node and ε is the ratio of the reduced Compton wavelength to the characteristic length of the system. The rule contains no free parameter and covers every bound state of every smooth potential. Two applications follow. For hydrogen it reproduces in three lines the closed coefficient obtained previously by radial integration. For the harmonic oscillator it gives a finite sum over the roots of the Hermite polynomials; the first two evaluate in closed form. The oscillator spectrum ceases to be equidistant, with two parameter-free level-spacing ratios, 1.4261 and 1.7514, which separate this model from the commutator branch, where the corresponding ratios are integers. The relative shift scales as the square root of the trap frequency. Two questions left open in the companion paper are settled: the momentum entering the deformed uncertainty relation is the one a time-of-flight measurement records, and the leading constant of that relation is 4, so that the Gaussian is its minimum-uncertainty state to first order. The same constant governs the entropy-power form, which is uniformly stronger for every non-Gaussian state; the corresponding statement for the product of entropy powers is formulated as a conjecture, with the obstruction to proving it identified. Throughout, other deformations are discussed under an explicit rule: parameters of different frameworks are never compared, only predictions for one and the same measured observable. The paper claims neither the family of equations nor the node mechanism as new, and reports no evidence for a nonzero deformation. Companion paper: https://doi.org/10.5281/zenodo.22072059A working log documenting all calculations and twenty-one self-corrections is included in the files.
Parwani and Tabia established in 2006 that in information-theoretic nonlinear extensions of the Schrödinger equation the energy shift of an excited state is carried predominantly by the nodes of the unperturbed wavefunction, while nodeless ground states receive only subleading shifts. This paper shows that for the rotation-invariant branch of that family, regulated by the small components of the Dirac spinor as proposed by Liew and Parwani, the node contribution closes in a single expression: each node of the density contributes exactly −4πC²/ε to the shift integral, where C is the slope of the unnormalised wavefunction at the node and ε is the ratio of the reduced Compton wavelength to the characteristic length of the system. The rule contains no free parameter and covers every bound state of every smooth potential. Two applications follow. For hydrogen it reproduces in three lines the closed coefficient obtained previously by radial integration. For the harmonic oscillator it gives a finite sum over the roots of the Hermite polynomials; the first two evaluate in closed form. The oscillator spectrum ceases to be equidistant, with two parameter-free level-spacing ratios, 1.4261 and 1.7514, which separate this model from the commutator branch, where the corresponding ratios are integers. The relative shift scales as the square root of the trap frequency. Two questions left open in the companion paper are settled: the momentum entering the deformed uncertainty relation is the one a time-of-flight measurement records, and the leading constant of that relation is 4, so that the Gaussian is its minimum-uncertainty state to first order. The same constant governs the entropy-power form, which is uniformly stronger for every non-Gaussian state; the corresponding statement for the product of entropy powers is formulated as a conjecture, with the obstruction to proving it identified. Throughout, other deformations are discussed under an explicit rule: parameters of different frameworks are never compared, only predictions for one and the same measured observable. The paper claims neither the family of equations nor the node mechanism as new, and reports no evidence for a nonzero deformation. Companion paper: https://doi.org/10.5281/zenodo.22072059A working log documenting all calculations and twenty-one self-corrections is included in the files.
The slowly varying envelope (SVE) reduction of a gapped classical electromagnetic carrier predicts the single-particle Schrödinger equation as an exact identity plus a closed-form residual: the constrained vector MacCullagh carrier admits an exact per-branch balance whose retained part is the Schrödinger operator and whose sole marked approximation leaves a closed-form remainder, quadratic in the detuning. Under the single identification below, the carrier branch ℏω(k) is the relativistic energy function E(p) = m2 c4 + c2 p2 and the envelope generator ℏ[ω(k) − ωC ] is p its kinetic part E(p) − mc2 , recovered consistently in three cross-checking registers—the Foldy– Wouthuysen operator, the dispersion Taylor series, and direct numerics—with leading correction the fine-structure term −p4 /8m3 c2 . A parameter-free model reproduces the archived residual slopes to 10−4 , the fitted exponent converging to the predicted 2. One action-scale identification, ℏωC = mc2 , enters exactly once as an input; by Buckingham-π the reduction fixes the coefficient c2 /ωC = ω ′′ (0), matched to ℏ/m under the identification, and leaves one free scale, so ℏ is supplied, not derived. The result is a dynamical correspondence at the envelope stratum, not a derivation of quantum mechanics: |ψ|2 is normalized wave action, never a Born probability, and an explicit ledger fences what the correspondence forces, permits, and does not produce—the Born rule, state reduction, quantized charge, and entanglement—each with its standing theorem or structural fact. Companion no-go paper: "Constraining Classical Field Models of Self-Generated Particle Structure: Successive No-Go Theorems", DOI 10.5281/zenodo.21531712 (all versions). The two papers are co-posted and cross-cite by concept DOI. This record additionally carries the complete Track S/N/P code-and-data archive (envelope-code-v3.35.zip): symbolic-audit, numerical, and magnitude pipelines with their test suites (106, 20, and 13 tests), all figure-generating scripts, the campaign manifests (track_n_campaign_summary.json and companions), and dependency floors (code/requirements.txt). Version 2 (September 2026). Retitled ("a Gapped Electromagnetic Carrier") and substantially revised; 49 → 31 pages. Principal changes since v1: (i) the reduction theorem now carries explicit hypotheses — ψ₀ ∈ H^{s+4} via a bounded-multiplier argument, with a band-limit alternative and the H^s-weighted packet functional K_{4,s} (eighth-moment form) replacing the earlier spectral-scale shorthand; (ii) the residual analysis adds the finite-sweep bias law with a parameter-free model reproducing all archived window slopes to 1e-4, guarded by a committed test file; (iii) Sections V and VI are merged and the four scope tables consolidated into a single honesty-ledger table with explicit class codes; (iv) the Wolff placement is rebuilt as a three-horn trilemma mirroring the companion paper; (v) the conclusion is rewritten, situating the result against the independent kinematic (Zhang) and intrinsic-periodicity (Puddu) routes; (vi) the effective-medium reading of the gapped carrier is stated as a constant-coefficient effective field theory with no substrate assumed or determined; (vii) the carrier branch and the envelope generator are distinguished throughout, the companion dependence is stated results-based, and the long-time figure marks the Gaussian-packet breakdown estimate; (viii) all printed record DOIs now use the Zenodo concept (all-versions) DOIs. A complete change ledger (paper1_delta_v3.13_to_v3.35.pdf) accompany this version, together with the complete Track S/N/P code and data archive (envelope-code-v3.35.zip).
The well-known Heisenberg–Robertson uncertainty relation for a pair of noncommuting observables, is expressed in terms of the product of variances and the commutator among the operators, computed for the quantum state of a system. Different modified commutation relations have been considered in the last years with the purpose of taking into account the effect of quantum gravity. Indeed it can be seen that letting [ X , P ] = i ℏ ( 1 + β P 2 ) implies the existence of a minimal length proportional to β . The Bialynicki-Birula–Mycielski entropic uncertainty relation in terms of Shannon entropies is also seen to be deformed in the presence of a minimal length, corresponding to a strictly positive deformation parameter β . Generalized entropies can be implemented. Indeed, results for the sum of position and (auxiliary) momentum Rényi entropies with conjugated indices have been provided recently for the ground and first excited state. We present numerical findings for conjugated pairs of entropic indices, for the lowest lying levels of the deformed harmonic oscillator system in 1D, taking into account the position distribution for the wavefunction and the actual momentum.
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.
The well-known Heisenberg--Robertson uncertainty relation for a pair of noncommuting observables, is expressed in terms of the product of variances and the commutator among the operators, computed for the quantum state of a system. Different modified commutation relations have been considered in the last years with the purpose of taking into account the effect of quantum gravity. Indeed it can be seen that letting [X,P]=iℏ(1+βP2) implies the existence of a minimal length proportional to β. The Bialynicki-Birula--Mycielski entropic uncertainty relation in terms of Shannon entropies is also seen to be deformed in the presence of a minimal length, corresponding to a strictly positive deformation parameter β. Generalized entropies can be implemented. Indeed, results for the sum of position and (auxiliary) momentum Rényi entropies with conjugated indices have been provided recently for the ground and first excited state. We present numerical findings for conjugated pairs of entropic indices, for the lowest lying levels of the deformed harmonic oscillator system in 1D, taking into account the position distribution for the wavefunction and the actual momentum.
Degeneracy allows the nodal structure of a quantum state to vary without changing its energy. We study this effect for real superpositions in the fixed-energy shells of the two-dimensional isotropic harmonic oscillator. In Cartesian coordinates (x,y), any such state in the Nth shell has the parametrized form ψN(x,y;c)=ψ0(x,y)PN(x,y;c), where ψ0 is the non-vanishing ground state and the real coefficients c determine the polynomial factor PN. The nodal set is exactly the algebraic curve PN=0. Thus, varying c reshapes this curve at fixed energy. We show that its topology can change only when it develops a finite singular point, PN=∇PN=0, or when two asymptotic nodal directions merge. We characterize this geometry using three diagnostics: the nodal-domain entropy Sdom, the Cartesian mutual information I(x;y), and the entropic uncertainty sum Sr+Sp, which respectively probe probability redistribution among nodal domains, coordinate correlations, and global position-momentum delocalization. The lowest shells reveal a clear hierarchy. For N=1, mixing only rotates a nodal line. Along a representative N=2 path, a closed conic passes through parallel lines into a hyperbola-type curve; Sdom clearly detects this transition, while Sr+Sp remains smooth. For N=3, merging asymptotic directions brings smooth cubic branches close together and enhances both Sdom and I(x;y). These signatures are accessible in Hermite-Gaussian structured light and nearly isotropic trapped oscillators.
In the context of a two-parameter (α, β) deformation of the canonical commutation relation leading to nonzero minimal uncertainties in both position and momentum, the harmonic oscillator spectrum and eigenvectors are determined by using an extension of the techniques of conventional supersymmetric quantum mechanics (SUSYQM) combined with shape invariance under parameter scaling. The resulting supersymmetric partner Hamiltonians correspond to different masses and frequencies. The exponential spectrum is proved to reduce to a previously found quadratic spectrum whenever one of the parameters α, β vanishes, in which case shape invariance under parameter translation occurs. In the special case where α = β ≠ 0, the oscillator Hamiltonian is shown to coincide with that of the q -deformed oscillator with q > 1 and its eigenvectors are therefore n - q -boson states. In the general case where 0 ≠ α ≠ β ≠ 0, the eigenvectors are constructed as linear combinations of n - q -boson states by resorting to a Bargmann representation of the latter and to q -differential calculus. They are finally expressed in terms of a q -exponential and little q -Jacobi polynomials.