Numerical Stability in Quantum Tunneling: The Role of the AMDCF Absorbing Boundary Kernel

The simulation of quantum tunneling using the Time-Dependent Schrödinger Equation (TDSE) on a finite computational grid is highly susceptible to artificial boundary reflections. When a wavepacket reaches the edge of the grid, periodic boundary conditions (inherent in the Fourier transform) cause the wave to re-enter from the opposite side, contaminating the physical results. In this study, we present a comparative analysis. Without the AMDCF kernel, the probability transfer plots exhibit severe oscillations and false tunneling events. However, by applying the AMDCF dynamic safety valve as a smooth absorbing boundary, the wavepacket is perfectly attenuated at the edges, resulting in a stable, physically accurate probability transfer (reflected: 85%, tunneled: 15%). This work demonstrates that the AMDCF framework is not only a control algorithm but also a robust mathematical tool for ensuring numerical stability in complex quantum simulations.