The Stimulated Einstein Branch of the Scretching Quantum Chain B₁₂, B₂₁, Degeneracy-Normalized Equality, and New SQC Invariants
Daniel Scretching
Description The Stimulated Einstein Branch of the Scretching Quantum Chain: B12,B21B_{12}, B_{21}B12,B21, Degeneracy-Normalized Equality, and New SQC Invariants extends the Scretching Quantum Chain (SQC) framework into the domain of stimulated radiative processes, completing the structural picture initiated in earlier work on spontaneous emission. This paper systematically incorporates the Einstein BBB-coefficients—governing absorption and stimulated emission—into the SQC formalism, demonstrating that they obey the same invariant structure already identified in the spontaneous AAA-sector. Building from the established hydrogenic SQC state vector, which includes transition dipole strength ∣μ12∣2|\mu_{12}|^2∣μ12∣2, oscillator strength f12f_{12}f12, and the Einstein coefficients A21,B21,B12A_{21}, B_{21}, B_{12}A21,B21,B12, the work derives two new stimulated-branch invariants. These are constructed by combining the previously published spontaneous-emission invariant with the standard Einstein reciprocity relations. The resulting quantities, IB,21=B21 ν(g1/g2)f12,IB,12=B12 νf12,I_{B,21} = \frac{B_{21}\,\nu}{(g_1/g_2)f_{12}}, \quad I_{B,12} = \frac{B_{12}\,\nu}{f_{12}},IB,21=(g1/g2)f12B21ν,IB,12=f12B12ν, are shown to collapse to a single constant, KB=e24ε0meh,K_B = \frac{e^2}{4\varepsilon_0 m_e h},KB=4ε0mehe2, revealing a unified invariant structure across both stimulated emission and absorption processes. A key conceptual result is the clarification of degeneracy effects: while the raw Einstein relation g1B12=g2B21g_1 B_{12} = g_2 B_{21}g1B12=g2B21 reflects intrinsic asymmetry, the paper shows that the stimulated coefficients become exactly equal when expressed in degeneracy-normalized form. This normalization exposes a deeper symmetry within the radiative system, aligning both coefficients with the same invariant constant. The framework is validated through hydrogen Lyman-series transitions, confirming numerical closure using previously published datasets. The analysis remains fully consistent with the standard radiative theory originally developed by Albert Einstein, and does not introduce new physical laws. Instead, it reorganizes known relationships into a deterministic invariant chain that unifies spontaneous emission, stimulated emission, and absorption within a single algebraic structure. Overall, this work completes the extension of the SQC formalism into the stimulated regime, demonstrating that all three radiative processes share a common invariant foundation. The result is a more comprehensive and internally consistent framework for understanding radiative transitions in hydrogenic systems.
•Feb 25, 2026•0 citations