Numerical Investigation of the Time-Dependent Schrödinger Equation Via Finite Difference Approach

Abstract

This paper presents a conceptually straightforward, accurate, and efficient numerical approach for obtaining solutions to the time-dependent Schrödinger equation (TDSE), utilizing the Finite Difference Method (FDM). The methodology employs a forward difference scheme for temporal derivatives and a second-order central difference scheme for spatial derivatives, with the computational framework implemented in MATLAB. The efficacy of the proposed method is rigorously demonstrated through simulations of an electron wave packet in various scenarios. Key findings include the confirmation of fundamental quantum principles, such as the Heisenberg Uncertainty Principle and the conservation of total energy (expectation value) throughout system evolution. Analysis of wave packet interactions with both attractive and repulsive step barriers reveal phenomena of dispersion, probabilistic reflection, and transmission, with the total probability consistently conserved. Furthermore, the dynamics of electron wave packets in uniform electric fields (accelerating, retarding, and zero-field conditions) are explored, illustrating energy transformations and providing insights into their behaviour in comparison to classical predictions. These simulations collectively underscore the robustness and versatility of the FDM approach, which is readily extensible to higher dimensions and diverse time-dependent applications, thereby contributing to a deeper computational understanding of quantum mechanical systems.