Full Derivation of the Scretching-Schrodinger Equation with Corrected Hydrogen Spectroscopic Closure

This paper derives the Scretching–Schrödinger Equation (SSE) in full theoretical detail from the ordinary Schrödinger equation, transition dipole theory, oscillator-strength formalism, Einstein coefficient theory, and the Maxwell–Scretching experimental closure. The derivation begins with the time-dependent Schrödinger equation, separates the stationary time-independent form, and then connects quantum-state evolution to spectroscopic observables through the transition dipole moment, oscillator strength, spontaneous-emission coefficient, and absorption-emission closure relations. Within this framework, the SQC spectroscopic invariant is introduced as

Kν=2πe2ε0mec3K_{\nu}=\frac{2\pi e^{2}}{\varepsilon_{0}m_{e}c^{3}}Kν=ε0mec32πe2,

which links electromagnetic coupling, electron mass, light speed, and frequency-dependent spectroscopic behavior.

The paper then derives the SSE closure constant,

KSSE=3ℏ(4.32×10−9)2meK_{\mathrm{SSE}}=\frac{3\hbar(4.32\times10^{-9})}{2m_{e}}KSSE=2me3ℏ(4.32×10−9),

as the coefficient that transfers the Maxwell–Scretching experimental closure into the quantum wave-equation formalism. The resulting Scretching–Schrödinger Equation is formulated as a constrained wave equation in which the Hamiltonian contains an additional spectroscopic correction operator, VSQCV_{\mathrm{SQC}}VSQC. This operator represents the SQC-imposed correction that connects ordinary quantum-mechanical evolution with experimentally recoverable spectroscopic closure behavior.

The derivation also corrects two important formal issues. First, the Hamiltonian is treated properly as an unbounded self-adjoint operator acting on a suitable dense domain in Hilbert space, rather than as an ordinary finite matrix or unrestricted algebraic operator. Second, the hydrogen $1s\to 2p$ radial transition integral is written in its corrected form,

276 a035\frac{2^{7}\sqrt{6}\,a_{0}}{3^{5}}35276a0,

rather than the incorrect expression

282 a035\frac{2^{8}\sqrt{2}\,a_{0}}{3^{5}}35282a0.

Together, these corrections strengthen the mathematical foundation of the SSE and ensure that the spectroscopic derivation remains consistent with standard quantum mechanics, hydrogenic transition theory, and the Maxwell–Scretching closure framework [1–5].