摘要
We employ the nonlinear quantum–classical transition equation and its equivalent scaled time-dependent Schrödinger equation to investigate wave-packet scattering from the Eckart well across classical, quantum, and superquantum regimes. A single control parameter, the degree of quantumness ϵ, continuously tunes the strength of the quantum contribution and provides a unified hydrodynamic and trajectory-based framework for analyzing scattering dynamics. By combining the exact energy-dependent transmission probability with time-dependent wave-packet propagation, we examine how reflection and transmission depend jointly on incident energy and quantumness. The results show that the scattering dynamics display broad near-transparent behavior in both the classical and superquantum limits, separated by an intermediate regime in which quantum reflection becomes appreciable, especially at low incident energies. In the classical limit, probability densities obtained from the scaled Schrödinger equation agree closely with those constructed from ensembles of classical trajectories, and the associated transition trajectories recover the expected classical motion. In addition, we show that the Eckart well becomes reflectionless at a discrete set of quantumness values, for which it reduces exactly to a reflectionless Pöschl–Teller potential and the transmission probability satisfies T(E)=1 for all incident energies. These results demonstrate that the transition-equation formalism provides a coherent description of classical, quantum, and superquantum scattering within a single theoretical setting. © 2026 Elsevier Inc.