A generalized finite-difference time-domain quantum method for the N-body interacting Hamiltonian

The Quantum Finite-Difference Time-Domain (FDTD-Q) method is a numerical method for solving the time evolution of the Schrödinger equation. It can be applied to systems of interacting particles, allowing for realistic simulations of quantum mechanics of various experimental systems. One of the drawbacks of the method is that divergences in the numerical evolution occur rather easily in the presence of interactions, which necessitates a large number of evolution steps or imaginary time evolution. We present a generalized (GFDTD-Q) method for solving the time-dependent Schrödinger equation including interactions between the particles. The new scheme provides a more relaxed condition for stability when the finite difference approximations for spatial derivatives are employed, as compared with the original FDTD-Q scheme. We demonstrate our scheme by simulating the time evolution of a two-particle interaction Hamiltonian. Our results show that the generalized method allows for stable time evolutions, in contrast to the original FDTD-Q scheme which produces a divergent solution.