<![CDATA[The time-dependent Schrödinger equation is fundamental to quantum mechanics, describing the temporal evolution of quantum systems. This research presents a High-Order Finite-Difference Time-Domain (HO-FDTD) method, employing Taylor series expansion to solve the equation with enhanced efficiency and accuracy. By advancing beyond traditional methods like first-order Taylor series (Crank-Nicolson, forward or backward Euler) or computationally intensive Runge-Kutta schemes, the HO-FDTD method leverages higher-order Taylor expansion for the time evolution operator while simultaneously refining the Laplacian operator. This dual improvement enhances precision, allowing for accurate modeling of complex quantum phenomena. Focusing on quantum tunneling, a critical process where electrons traverse potential barriers despite insufficient classical energy, the study examines tunneling probabilities and electron behavior across barriers of varying thickness in semiconductors. The simulations reveal that thicker barriers reduce tunneling probabilities, amplify deviations in electron positions, and indicate energy transfer during interactions, with increased resistance lowering kinetic energy and raising potential energy. These findings emphasize the significant influence of barrier thickness on quantum tunneling and highlight the HO-FDTD method"™s capability to capture intricate quantum dynamics, establishing it as a robust tool for advancing research and applications in quantum mechanics.]]>
