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 S_E = f(w), with A = S_E / ħ. The original eleven-point parameter sweep gives a monotonic nonlinear locus when A is plotted against the center broadening Δw_rms. A new matched-state stress test independently changes longitudinal energy or transverse confinement and retunes the barrier to reproduce Δw_rms = 0.08, 0.18, and 0.35. The corresponding WKB actions spread by 50.2%, 31.1%, and 22.7% relative to the baseline; the exact-scattering exponent agrees, and a refined-grid calculation is stable. Thus a universal one-variable proxy A = F(Δw_rms) fails without refitting. This result does not falsify an otherwise unspecified microscopic relation S_E = f(w), because Δw_rms is an observable moment rather than the coordinate or tunneling path w. It shows that broadening alone is insufficient and that a richer state description or path-level definition is required. The work is non-peer-reviewed and does not claim evidence that nature has an accessible fourth spatial dimension. The supplement contains the calculation scripts, CSV data, numerical metadata, convergence checks, and figures; AI assistance is disclosed in the manuscript.