This paper clarifies the limits of representing spontaneous emission with a classical electromagnetic field sourced only by an atom’s mean dipole. Extending a pure-state hydrogen result to an arbitrary two-level density matrix, it separates total radiated power from the portion attributable to optical coherence. Under the electric-dipole, rotating-wave, Markov, and free-space approximations, positivity of the density matrix places a strict upper bound on the coherent fraction of the emitted intensity. The remainder is necessarily associated with quantum dipole fluctuations rather than the classical mean field. A hydrogen benchmark and ensemble structure-factor analysis illustrate how excited-state population, coherence, and radiation can be measured and tested independently. The result provides a diagnostic boundary for semiclassical Maxwell–Scretching Chain (MSC) and Scretching Quantum Chain (SQC) source models while remaining consistent with established spontaneous-emission theory rather than proposing a new emission law.
This paper clarifies the limits of representing spontaneous emission with a classical electromagnetic field sourced only by an atom’s mean dipole. Extending a pure-state hydrogen result to an arbitrary two-level density matrix, it separates total radiated power from the portion attributable to optical coherence. Under the electric-dipole, rotating-wave, Markov, and free-space approximations, positivity of the density matrix places a strict upper bound on the coherent fraction of the emitted intensity. The remainder is necessarily associated with quantum dipole fluctuations rather than the classical mean field. A hydrogen benchmark and ensemble structure-factor analysis illustrate how excited-state population, coherence, and radiation can be measured and tested independently. The result provides a diagnostic boundary for semiclassical Maxwell–Scretching Chain (MSC) and Scretching Quantum Chain (SQC) source models while remaining consistent with established spontaneous-emission theory rather than proposing a new emission law.
This paper verifies the Maxwell–Scretching Chain (MSC) and the Scretching Quantum Chain (SQC) as closed mathematical transformation chains. The MSC begins with measured absorbance and derives the linked optical quantities Aλ, ελ, α_abs, κ, εᵣ″, σ, and f_eff. The SQC begins with frequency, wavelength, oscillator strength, and transition dipole strength, then derives photon energy, |μ₂₁|², A₂₁, and τ. The hydrogen atom is used as the control example because it provides a fully known physical system in which the major links of the chain can be checked directly. Maxwell electrodynamics supplies the Coulomb field of the proton, Schrödinger wave mechanics supplies the bound-state eigenfunctions and quantized energy levels, and the Planck–Einstein relation supplies the emitted-photon condition: ΔE = ℏω. When these components are combined, the final chain recovers the Bohr radii, Schrödinger hydrogen energy levels, Rydberg spectral wavelengths, and standard spontaneous-emission scaling. The result is a verified derived framework that organizes known physics into a closed transformation chain. It is not presented as an independent replacement for Maxwell electrodynamics, Schrödinger wave mechanics, Dirac theory, or quantum electrodynamics, but as a structured bridge connecting classical optics, quantum transition theory, and hydrogen spectral validation.
We derive the form of the one-photon state of the electromagnetic field resulting from spontaneous optical emission of a randomly oriented chiral quantum source, including electric dipole, magnetic dipole, and electric quadrupole contributions to the radiation. We describe how spatial coherence in the emission can be exploited to reveal hidden information about the emitter. This work serves as a prelude and companion piece to our concurrently submitted work \cite{Backlund2026Letter} in which we present the classical and quantum error bounds associated with assigning the handedness of such an emitter.
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.
Spontaneous emission in dipole approximation is studied theoretically using both source-field theory and a Schrodinger picture approach. Using source-field theory we obtain formal equations for the Poynting vector and energy density without making the rotating wave approximation (RWA) and Weisskopf-Wigner approximation (WWA). The initial condition at t=0 is one in which the atom is in an excited state and the field in the vacuum state. The source-field expressions are evaluated within the the RWA and WWA and are found to satisfy Poynting's theorem. To explore the consequences of not making the RWA and WWA, the Poynting vector and energy density are calculated using perturbation theory. We use a Schrodinger picture approach and essentially reproduce and complement the results of Compagno, Passante, and Persico [J. Mod. Optics 37:8, 1377 (2007)] and those of Power and Thirunamachandran [Phys. Rev. A 45, 54 (1992)] obtained using a Heisenberg picture approach. The theory involves a sum over field mode frequencies and both finite cutoffs and convergence factors are used to carry out the sums. It is shown that the perturbation theory calculation leads to unphysical values for atomic state populations for all times when a sum over all field frequencies is taken, even if a convergence factor is used. It is also proved that the fields calculated using source-field theory always satisfy Poynting's theorem for ct not equal to R, where R is the distance from the atom.
Starting from a quantum Maxwell equation for the vector potential operator, we study the spontaneous emission properties of a two-level atom interacting with the electromagnetic field in a lossless and inhomogeneous dielectric structure and express the spontaneous emission rate, the external quantum efficiency, and the spontaneous emission factor in terms of the classical Green function of the dielectric structure. Comparing these results with the corresponding classical results, we show that the above quantities can be calculated from the radiation field of a classical dipole. Based on this correspondence, a numerical algorithm is developed to calculate the modification of spontaneous emission in microcavities. The line shape of the emission spectrum of the light source and the decay of the cavity resonant modes are automatically taken into account in this algorithm. The approximate expressions for the spontaneous emission rate and the spontaneous emission factor are discussed.
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].