Coupled-Channel Quantum Tunneling with an Additional Confined Coordinate

Abstract

This exploratory preprint studies a controlled quantum-scattering toy model with an additional harmonically confined coordinate w. It compares a fixed-width one-dimensional barrier, the exact shrinking observable slice V(x,0), and the full coupled (x,w) Hamiltonian. The identical-slice control shows that the non-monotonic transmission turn is not unique to an additional coordinate, while the coupled calculation produces a quantitatively distinct transmission curve and a converged transverse-broadening signal. The action-coordinate hypothesis is formulated as S_E=f(w), with A=S_E/hbar and K(w)=dA/dw. When the eleven numerical samples are ordered by the solver-derived center RMS broadening, A_WKB is strictly increasing and is therefore compatible with a nonlinear empirical proxy A=F(Delta w_rms). The constant-K linear subcase fails globally, but this does not reject the general S_E=f(w) hypothesis. Because Delta w_rms is an observable proxy rather than the coordinate w itself, and because only one prescribed parameter family is tested, the calculation does not identify a unique or universal f. The work is explicitly non-peer-reviewed and does not claim evidence that nature has an accessible fourth spatial dimension. The accompanying supplement contains the complete Python calculation, CSV results, numerical metadata, convergence checks, and figures. AI assistance is disclosed in the manuscript.