Conservation properties of a leapfrog finite-difference time-domain method for the Schrödinger equation

We study the probability and energy conservation properties of a leap-frog finite-difference time-domain (FDTD) method for solving the Schr\"odinger equation. We propose expressions for the total numerical probability and energy contained in a region, and for the flux of probability current and power through its boundary. We show that the proposed expressions satisfy the conservation of probability and energy under suitable conditions. We demonstrate their connection to the Courant-Friedrichs-Lewy condition for stability. We argue that these findings can be used for developing a modular framework for stability analysis in advanced algorithms based on FDTD for solving the Schr\"odinger equation.